A&G-Introduction to algebra

Linear Equations - All Forms

LINEAR EQUATIONS - All Forms

📘 Linear Equations - Key Concepts

A linear equation is an equation where the highest power of the variable is 1. They can be written in several forms.

Three Main Forms:

Gradient-Intercept Form: y = mx + c
Point-Gradient Form: y - y₁ = m(x - x₁)
General Form: ax + by + c = 0

Example 1 (Solve for x): 3x + 5 = 20

3x + 5 = 20
3x = 20 - 5 = 15
x = 15 ÷ 3 = 5

Example 2 (With brackets): 2(x + 3) = 14

2x + 6 = 14
2x = 14 - 6 = 8
x = 8 ÷ 2 = 4

Example 3 (With fractions): (x/2) + 3 = 7

x/2 = 7 - 3 = 4
x = 4 × 2 = 8

📌 Section A: 10 Linear Expressions (Simplify Only)

Instructions: Simplify each linear expression by collecting like terms.

These are expressions, not equations - no solving required.

1. 3x + 5x - 2x

2. 7x - 3x + 4

3. 2(3x + 4) - 5x

4. 4(x + 2) + 3(x - 1)

5. 5x - 2(3x - 4)

6. 3(2x + 1) + 2(4x - 3)

7. 6x - (2x + 5) + 3

8. 4(2x - 3) - 3(5x + 2)

9. 2x + 3 + 5x - 7 + 4x

10. 8 - 3(2x - 1) + 4x

✅ Solutions - Section A

1. 3x + 5x - 2x = 6x

2. 7x - 3x + 4 = 4x + 4

3. 2(3x + 4) - 5x = 6x + 8 - 5x = x + 8

4. 4(x + 2) + 3(x - 1) = 4x + 8 + 3x - 3 = 7x + 5

5. 5x - 2(3x - 4) = 5x - 6x + 8 = -x + 8

6. 3(2x + 1) + 2(4x - 3) = 6x + 3 + 8x - 6 = 14x - 3

7. 6x - (2x + 5) + 3 = 6x - 2x - 5 + 3 = 4x - 2

8. 4(2x - 3) - 3(5x + 2) = 8x - 12 - 15x - 6 = -7x - 18

9. 2x + 3 + 5x - 7 + 4x = (2x + 5x + 4x) + (3 - 7) = 11x - 4

10. 8 - 3(2x - 1) + 4x = 8 - 6x + 3 + 4x = 11 - 2x

📌 Section B: 5 Simple Linear Equations (ax + b = c)

Instructions: Solve each linear equation for x.

These are simple equations in the form ax + b = c.

11. 4x + 7 = 19

12. 3x - 5 = 16

13. 5x + 3 = 23

14. 2x - 8 = 10

15. 6x + 4 = 34

✅ Solutions - Section B

11. 4x + 7 = 19
4x = 19 - 7 = 12
x = 12 ÷ 4 = 3

12. 3x - 5 = 16
3x = 16 + 5 = 21
x = 21 ÷ 3 = 7

13. 5x + 3 = 23
5x = 23 - 3 = 20
x = 20 ÷ 5 = 4

14. 2x - 8 = 10
2x = 10 + 8 = 18
x = 18 ÷ 2 = 9

15. 6x + 4 = 34
6x = 34 - 4 = 30
x = 30 ÷ 6 = 5

📌 Section C: 5 Equations with Brackets

Instructions: Solve each linear equation. First expand any brackets, then collect like terms and solve for x.

16. 2(x + 4) = 18

17. 3(2x - 1) = 21

18. 4(x + 3) + 5 = 29

19. 5(2x - 3) - 4 = 31

20. 2(3x + 1) + 3(2x - 1) = 23

✅ Solutions - Section C

16. 2(x + 4) = 18
2x + 8 = 18
2x = 18 - 8 = 10
x = 10 ÷ 2 = 5

17. 3(2x - 1) = 21
6x - 3 = 21
6x = 21 + 3 = 24
x = 24 ÷ 6 = 4

18. 4(x + 3) + 5 = 29
4x + 12 + 5 = 29
4x + 17 = 29
4x = 29 - 17 = 12
x = 12 ÷ 4 = 3

19. 5(2x - 3) - 4 = 31
10x - 15 - 4 = 31
10x - 19 = 31
10x = 31 + 19 = 50
x = 50 ÷ 10 = 5

20. 2(3x + 1) + 3(2x - 1) = 23
6x + 2 + 6x - 3 = 23
12x - 1 = 23
12x = 23 + 1 = 24
x = 24 ÷ 12 = 2

📌 Section D: 5 Equations with Fractions

Instructions: Solve each linear equation involving fractions. Multiply through by the common denominator to eliminate fractions first.

21. x/3 + 2 = 5

22. 2x/5 - 1 = 3

23. (x + 2)/4 = 3

24. (2x - 1)/3 = 5

25. x/2 + x/3 = 5

✅ Solutions - Section D

21. x/3 + 2 = 5
x/3 = 5 - 2 = 3
x = 3 × 3 = 9

22. 2x/5 - 1 = 3
2x/5 = 3 + 1 = 4
2x = 4 × 5 = 20
x = 20 ÷ 2 = 10

23. (x + 2)/4 = 3
x + 2 = 3 × 4 = 12
x = 12 - 2 = 10

24. (2x - 1)/3 = 5
2x - 1 = 5 × 3 = 15
2x = 15 + 1 = 16
x = 16 ÷ 2 = 8

25. x/2 + x/3 = 5
Common denominator = 6
(3x + 2x)/6 = 5
5x/6 = 5
5x = 5 × 6 = 30
x = 30 ÷ 5 = 6

📌 Section E: 5 Equations with x on Both Sides

Instructions: Solve each linear equation where x appears on both sides. Collect x terms on one side and constants on the other.

26. 5x + 3 = 2x + 12

27. 7x - 4 = 3x + 16

28. 4x + 7 = 9x - 8

29. 6x - 5 = 2x + 11

30. 3x + 8 = 5x - 4

✅ Solutions - Section E

26. 5x + 3 = 2x + 12
5x - 2x = 12 - 3
3x = 9
x = 9 ÷ 3 = 3

27. 7x - 4 = 3x + 16
7x - 3x = 16 + 4
4x = 20
x = 20 ÷ 4 = 5

28. 4x + 7 = 9x - 8
4x - 9x = -8 - 7
-5x = -15
x = (-15) ÷ (-5) = 3

29. 6x - 5 = 2x + 11
6x - 2x = 11 + 5
4x = 16
x = 16 ÷ 4 = 4

30. 3x + 8 = 5x - 4
3x - 5x = -4 - 8
-2x = -12
x = (-12) ÷ (-2) = 6

📌 Section F: 5 Equations with Negative Coefficients

Instructions: Solve each linear equation involving negative coefficients. Be careful with signs when moving terms.

31. -3x + 7 = 1

32. 5 - 2x = 11

33. 4 - 3x = 7 - 5x

34. 2(3 - x) = 10

35. 8 - 4x = 2x - 10

✅ Solutions - Section F

31. -3x + 7 = 1
-3x = 1 - 7 = -6
x = (-6) ÷ (-3) = 2

32. 5 - 2x = 11
-2x = 11 - 5 = 6
x = 6 ÷ (-2) = -3

33. 4 - 3x = 7 - 5x
-3x + 5x = 7 - 4
2x = 3
x = 1.5 or ³⁄₂

34. 2(3 - x) = 10
6 - 2x = 10
-2x = 10 - 6 = 4
x = 4 ÷ (-2) = -2

35. 8 - 4x = 2x - 10
-4x - 2x = -10 - 8
-6x = -18
x = (-18) ÷ (-6) = 3

📌 Section G: 5 Equations with Decimals

Instructions: Solve each linear equation involving decimals. You may multiply by a power of 10 to eliminate decimals if preferred.

36. 0.5x + 2.5 = 5.5

37. 1.2x - 3.4 = 5.0

38. 0.3(x + 4) = 2.1

39. 2.5x + 1.5 = 3.5x - 2.5

40. 0.75x - 2.25 = 1.5

✅ Solutions - Section G

36. 0.5x + 2.5 = 5.5
0.5x = 5.5 - 2.5 = 3.0
x = 3.0 ÷ 0.5 = 6

37. 1.2x - 3.4 = 5.0
1.2x = 5.0 + 3.4 = 8.4
x = 8.4 ÷ 1.2 = 7

38. 0.3(x + 4) = 2.1
0.3x + 1.2 = 2.1
0.3x = 2.1 - 1.2 = 0.9
x = 0.9 ÷ 0.3 = 3

39. 2.5x + 1.5 = 3.5x - 2.5
2.5x - 3.5x = -2.5 - 1.5
-1.0x = -4.0
x = (-4.0) ÷ (-1.0) = 4

40. 0.75x - 2.25 = 1.5
0.75x = 1.5 + 2.25 = 3.75
x = 3.75 ÷ 0.75 = 5

📊 Quick Reference Summary

Section	Type	Example	Key Step
A	Expressions	3x + 5x - 2x	Collect like terms
B	ax + b = c	4x + 7 = 19	Isolate x term, then divide
C	With brackets	2(x + 4) = 18	Expand brackets first
D	With fractions	x/3 + 2 = 5	Multiply by denominator
E	x on both sides	5x + 3 = 2x + 12	Collect x terms together
F	Negative coefficients	-3x + 7 = 1	Watch signs carefully
G	Decimals	0.5x + 2.5 = 5.5	Multiply by 10 if needed

📝 Key Takeaways for Linear Equations:

Simplify expressions by collecting like terms
Expand brackets before solving equations
For fractions: multiply through by the common denominator
For x on both sides: move all x terms to one side, constants to the other
Check your answer by substituting back into the original equation
Be careful with signs when moving terms across the equals sign
For decimals: multiply by 10, 100, etc. to work with whole numbers

End of Linear Equations Practice Problems

Linear Equations - Brackets & Multiple Letters

LINEAR EQUATIONS - Brackets & Multiple Letters

📘 Linear Equations with Multiple Variables

When solving linear equations with multiple letters (variables), we treat all variables as we would with numbers, but we may need to solve for one variable in terms of others.

Key Techniques:

Expand brackets using distributive property: a(b + c) = ab + ac
Collect like terms - terms with the same variable
Factor out the variable you're solving for
Isolate the variable by dividing

Example 1 (Solve for x): a(x + b) = c

ax + ab = c
ax = c - ab
x = (c - ab)/a

Example 2 (Solve for y): p(y - q) = r(y + s)

py - pq = ry + rs
py - ry = rs + pq
y(p - r) = rs + pq
y = (rs + pq)/(p - r)

Example 3 (Solve for a): 2(a + 3b) = 4a - b

2a + 6b = 4a - b
2a - 4a = -b - 6b
-2a = -7b
a = 7b/2

📌 Section A: 10 Expressions with Brackets (Simplify Only)

Instructions: Expand the brackets and simplify each expression by collecting like terms.

These are expressions, not equations - no solving required.

1. 2(x + 3y) + 4x - y

2. 3(2a - b) + 5(a + 2b)

3. 4(p + 2q) - 3(2p - q)

4. 5(2m + n) - 2(m - 3n) + 4m

5. 3(4x - 2y) + 2(3x + 5y) - x

6. a(2b + 3c) + b(4a - c) - 3ac

7. 2x(3y - z) + 4y(x + 2z) - 3xz

8. 3p(2q + r) - 2q(p - 3r) + pr

9. 5(2x + 3y - z) - 3(4x - y + 2z)

10. 2a(3b - 4c) + 3b(2a + 5c) - 4c(3a - 2b)

✅ Solutions - Section A

1. 2(x + 3y) + 4x - y
= 2x + 6y + 4x - y
= (2x + 4x) + (6y - y)
= 6x + 5y

2. 3(2a - b) + 5(a + 2b)
= 6a - 3b + 5a + 10b
= (6a + 5a) + (-3b + 10b)
= 11a + 7b

3. 4(p + 2q) - 3(2p - q)
= 4p + 8q - 6p + 3q
= (4p - 6p) + (8q + 3q)
= -2p + 11q

4. 5(2m + n) - 2(m - 3n) + 4m
= 10m + 5n - 2m + 6n + 4m
= (10m - 2m + 4m) + (5n + 6n)
= 12m + 11n

5. 3(4x - 2y) + 2(3x + 5y) - x
= 12x - 6y + 6x + 10y - x
= (12x + 6x - x) + (-6y + 10y)
= 17x + 4y

6. a(2b + 3c) + b(4a - c) - 3ac
= 2ab + 3ac + 4ab - bc - 3ac
= (2ab + 4ab) + (3ac - 3ac) - bc
= 6ab - bc

7. 2x(3y - z) + 4y(x + 2z) - 3xz
= 6xy - 2xz + 4xy + 8yz - 3xz
= (6xy + 4xy) + (-2xz - 3xz) + 8yz
= 10xy - 5xz + 8yz

8. 3p(2q + r) - 2q(p - 3r) + pr
= 6pq + 3pr - 2pq + 6qr + pr
= (6pq - 2pq) + (3pr + pr) + 6qr
= 4pq + 4pr + 6qr

9. 5(2x + 3y - z) - 3(4x - y + 2z)
= 10x + 15y - 5z - 12x + 3y - 6z
= (10x - 12x) + (15y + 3y) + (-5z - 6z)
= -2x + 18y - 11z

10. 2a(3b - 4c) + 3b(2a + 5c) - 4c(3a - 2b)
= 6ab - 8ac + 6ab + 15bc - 12ac + 8bc
= (6ab + 6ab) + (-8ac - 12ac) + (15bc + 8bc)
= 12ab - 20ac + 23bc

📌 Section B: 5 Equations - Solve for x in terms of other letters

Instructions: Solve each equation for x. Express your answer in terms of the other variables.

11. a(x + b) = c

12. p(x - q) = r

13. m(x + n) = k(x + p)

14. 2(ax + b) = 3(cx - d)

15. 5(2x - 3y) = 4(3x + 2y)

✅ Solutions - Section B

11. a(x + b) = c
ax + ab = c
ax = c - ab
x = (c - ab)/a

12. p(x - q) = r
px - pq = r
px = r + pq
x = (r + pq)/p

13. m(x + n) = k(x + p)
mx + mn = kx + kp
mx - kx = kp - mn
x(m - k) = kp - mn
x = (kp - mn)/(m - k)

14. 2(ax + b) = 3(cx - d)
2ax + 2b = 3cx - 3d
2ax - 3cx = -3d - 2b
x(2a - 3c) = -(3d + 2b)
x = -(3d + 2b)/(2a - 3c)

15. 5(2x - 3y) = 4(3x + 2y)
10x - 15y = 12x + 8y
10x - 12x = 8y + 15y
-2x = 23y
x = -23y/2

📌 Section C: 5 Equations - Solve for y in terms of x and others

Instructions: Solve each equation for y. Express your answer in terms of x and any other variables.

16. 3x + 2y = 12

17. ax + by = c

18. p(2x + y) = q(x - y)

19. 4(x - 2y) = 3(2x + y) - 5

20. m(x + 3y) = n(2x - y) + kx

✅ Solutions - Section C

16. 3x + 2y = 12
2y = 12 - 3x
y = (12 - 3x)/2

17. ax + by = c
by = c - ax
y = (c - ax)/b

18. p(2x + y) = q(x - y)
2px + py = qx - qy
py + qy = qx - 2px
y(p + q) = x(q - 2p)
y = x(q - 2p)/(p + q)

19. 4(x - 2y) = 3(2x + y) - 5
4x - 8y = 6x + 3y - 5
-8y - 3y = 6x - 4x - 5
-11y = 2x - 5
y = (5 - 2x)/11

20. m(x + 3y) = n(2x - y) + kx
mx + 3my = 2nx - ny + kx
3my + ny = 2nx + kx - mx
y(3m + n) = x(2n + k - m)
y = x(2n + k - m)/(3m + n)

📌 Section D: 5 Equations with Three Variables

Instructions: Solve each equation for the indicated variable.

21. Solve for x: 2x + 3y - 4z = 10

22. Solve for a: 3a - 2b + 5c = 12

23. Solve for p: 4(p - 2q) = 3(r + 2p)

24. Solve for m: 2(3m - n) = 5(2m + p) - 3n

25. Solve for u: 3(u + 2v) - 2(3u - w) = 4(v - w)

✅ Solutions - Section D

21. 2x + 3y - 4z = 10
2x = 10 - 3y + 4z
x = (10 - 3y + 4z)/2

22. 3a - 2b + 5c = 12
3a = 12 + 2b - 5c
a = (12 + 2b - 5c)/3

23. 4(p - 2q) = 3(r + 2p)
4p - 8q = 3r + 6p
4p - 6p = 3r + 8q
-2p = 3r + 8q
p = -(3r + 8q)/2

24. 2(3m - n) = 5(2m + p) - 3n
6m - 2n = 10m + 5p - 3n
6m - 10m = 5p - 3n + 2n
-4m = 5p - n
m = -(5p - n)/4

25. 3(u + 2v) - 2(3u - w) = 4(v - w)
3u + 6v - 6u + 2w = 4v - 4w
-3u + 6v + 2w = 4v - 4w
-3u = 4v - 4w - 6v - 2w
-3u = -2v - 6w
3u = 2v + 6w
u = (2v + 6w)/3

📌 Section E: 5 Equations with Fractions and Letters

Instructions: Solve each equation for the indicated variable. Clear fractions first by multiplying by the common denominator.

26. Solve for x: (x/a) + (y/b) = 1

27. Solve for y: (2x)/3 + y/4 = c

28. Solve for a: (a + b)/c = d

29. Solve for p: (p - q)/r = (p + s)/t

30. Solve for m: (m + n)/2 + (m - n)/3 = k

✅ Solutions - Section E

26. (x/a) + (y/b) = 1
Multiply by ab: bx + ay = ab
bx = ab - ay
x = (ab - ay)/b

27. (2x)/3 + y/4 = c
Multiply by 12: 8x + 3y = 12c
3y = 12c - 8x
y = (12c - 8x)/3

28. (a + b)/c = d
a + b = cd
a = cd - b

29. (p - q)/r = (p + s)/t
Cross multiply: t(p - q) = r(p + s)
tp - tq = rp + rs
tp - rp = rs + tq
p(t - r) = rs + tq
p = (rs + tq)/(t - r)

30. (m + n)/2 + (m - n)/3 = k
Multiply by 6: 3(m + n) + 2(m - n) = 6k
3m + 3n + 2m - 2n = 6k
5m + n = 6k
5m = 6k - n
m = (6k - n)/5

📊 Quick Reference Summary

Section	Type	Example	Key Step
A	Simplify expressions	2(x + 3y) + 4x - y	Expand brackets, collect like terms
B	Solve for x	a(x + b) = c	Expand, isolate x terms, factor if needed
C	Solve for y	3x + 2y = 12	Move x term, divide by coefficient of y
D	Three variables	2x + 3y - 4z = 10	Isolate the variable you're solving for
E	Fractions	(x/a) + (y/b) = 1	Multiply by common denominator

📝 Key Takeaways for Linear Equations with Multiple Letters:

Treat letters as numbers - the same algebraic rules apply
Expand brackets first using the distributive property
Collect like terms - terms with the same variables together
Factor out the variable you're solving for when it appears in multiple terms
For fractions: multiply through by the LCD to eliminate denominators
Check your answer by substituting back into the original equation
The final answer will be an expression containing other variables, not a number

End of Linear Equations - Brackets & Multiple Letters

Factorising Linear Expressions - Grouping Pairs

FACTORISING LINEAR EXPRESSIONS - Grouping Pairs

📘 Factorising by Grouping

When an expression has four terms like ax + bx + ay + by, we can factorise by grouping into two pairs.

Steps for Factorising by Grouping:

Group the terms into two pairs
Factor out the common factor from each pair
Identify the common bracket that appears in both groups
Factor out the common bracket

Example 1: ax + bx + ay + by

Group: (ax + bx) + (ay + by)
Factor each group: x(a + b) + y(a + b)
Common bracket: (a + b)
Factor out: (a + b)(x + y)

Example 2: 2x + 3y + 4x + 6y

Group: (2x + 4x) + (3y + 6y)
Factor: 2x(1 + 2) + 3y(1 + 2) ❌ This is better grouped by like terms first
Better grouping: (2x + 3y) + (4x + 6y)
Factor first group: (2x + 3y)
Factor second group: 2(2x + 3y)
Result: (2x + 3y)(1 + 2) = (2x + 3y)(3)

Example 3: 3a + 2b + 6a + 4b

Group: (3a + 2b) + (6a + 4b)
Factor first group: (3a + 2b)
Factor second group: 2(3a + 2b)
Result: (3a + 2b)(1 + 2) = (3a + 2b)(3)

Example 4: 2x - 3y + 4x - 6y

Group: (2x - 3y) + (4x - 6y)
Factor first group: (2x - 3y)
Factor second group: 2(2x - 3y)
Result: (2x - 3y)(1 + 2) = (2x - 3y)(3)

📌 Section A: 10 Basic Grouping Expressions

Instructions: Factorise each expression by grouping pairs. Look for a common bracket after grouping.

1. ax + bx + ay + by

2. px + qx + py + qy

3. 2a + 3b + 4a + 6b

4. 3x + 2y + 6x + 4y

5. 5m + 2n + 10m + 4n

6. 4p + 3q + 8p + 6q

7. 2x + 5y + 4x + 10y

8. 3a + 4b + 9a + 12b

9. 6x + 2y + 3x + y

10. 4m + 6n + 2m + 3n

✅ Solutions - Section A

1. ax + bx + ay + by
Group: (ax + bx) + (ay + by)
Factor: x(a + b) + y(a + b)
= (a + b)(x + y)

2. px + qx + py + qy
Group: (px + qx) + (py + qy)
Factor: x(p + q) + y(p + q)
= (p + q)(x + y)

3. 2a + 3b + 4a + 6b
Group: (2a + 3b) + (4a + 6b)
Factor first group: (2a + 3b)
Factor second group: 2(2a + 3b)
= (2a + 3b)(1 + 2) = (2a + 3b)(3) or 3(2a + 3b)

4. 3x + 2y + 6x + 4y
Group: (3x + 2y) + (6x + 4y)
Factor first group: (3x + 2y)
Factor second group: 2(3x + 2y)
= (3x + 2y)(1 + 2) = (3x + 2y)(3) or 3(3x + 2y)

5. 5m + 2n + 10m + 4n
Group: (5m + 2n) + (10m + 4n)
Factor first group: (5m + 2n)
Factor second group: 2(5m + 2n)
= (5m + 2n)(1 + 2) = (5m + 2n)(3) or 3(5m + 2n)

6. 4p + 3q + 8p + 6q
Group: (4p + 3q) + (8p + 6q)
Factor first group: (4p + 3q)
Factor second group: 2(4p + 3q)
= (4p + 3q)(1 + 2) = (4p + 3q)(3) or 3(4p + 3q)

7. 2x + 5y + 4x + 10y
Group: (2x + 5y) + (4x + 10y)
Factor first group: (2x + 5y)
Factor second group: 2(2x + 5y)
= (2x + 5y)(1 + 2) = (2x + 5y)(3) or 3(2x + 5y)

8. 3a + 4b + 9a + 12b
Group: (3a + 4b) + (9a + 12b)
Factor first group: (3a + 4b)
Factor second group: 3(3a + 4b)
= (3a + 4b)(1 + 3) = (3a + 4b)(4) or 4(3a + 4b)

9. 6x + 2y + 3x + y
Group: (6x + 2y) + (3x + y)
Factor first group: 2(3x + y)
Factor second group: (3x + y)
= (3x + y)(2 + 1) = (3x + y)(3) or 3(3x + y)

10. 4m + 6n + 2m + 3n
Group: (4m + 6n) + (2m + 3n)
Factor first group: 2(2m + 3n)
Factor second group: (2m + 3n)
= (2m + 3n)(2 + 1) = (2m + 3n)(3) or 3(2m + 3n)

📌 Section B: 10 Mixed Letters Grouping

Instructions: Factorise each expression by grouping. The common bracket may contain two different variables.

11. 2ax + 3bx + 2ay + 3by

12. 4px - 2qx + 4py - 2qy

13. 5mx + 5nx - 3my - 3ny

14. 3ax - 2bx - 3ay + 2by

15. 2ap + 3bp + 4aq + 6bq

16. 6x - 9y + 4xz - 6yz

17. 8ab + 12ac + 6b + 9c

18. 10xy - 15xz + 6y - 9z

19. 14mn + 21mp + 6n + 9p

20. 12pq - 18pr - 8q + 12r

✅ Solutions - Section B

11. 2ax + 3bx + 2ay + 3by
Group: (2ax + 3bx) + (2ay + 3by)
Factor first group: x(2a + 3b)
Factor second group: y(2a + 3b)
= (2a + 3b)(x + y)

12. 4px - 2qx + 4py - 2qy
Group: (4px - 2qx) + (4py - 2qy)
Factor first group: 2x(2p - q)
Factor second group: 2y(2p - q)
= 2(2p - q)(x + y) = 2(2p - q)(x + y)

13. 5mx + 5nx - 3my - 3ny
Group: (5mx + 5nx) + (-3my - 3ny)
Factor first group: 5x(m + n)
Factor second group: -3y(m + n)
= (m + n)(5x - 3y)

14. 3ax - 2bx - 3ay + 2by
Group: (3ax - 2bx) + (-3ay + 2by)
Factor first group: x(3a - 2b)
Factor second group: -y(3a - 2b)
= (3a - 2b)(x - y)

15. 2ap + 3bp + 4aq + 6bq
Group: (2ap + 3bp) + (4aq + 6bq)
Factor first group: p(2a + 3b)
Factor second group: 2q(2a + 3b)
= (2a + 3b)(p + 2q)

16. 6x - 9y + 4xz - 6yz
Group: (6x - 9y) + (4xz - 6yz)
Factor first group: 3(2x - 3y)
Factor second group: 2z(2x - 3y)
= (2x - 3y)(3 + 2z)

17. 8ab + 12ac + 6b + 9c
Group: (8ab + 12ac) + (6b + 9c)
Factor first group: 4a(2b + 3c)
Factor second group: 3(2b + 3c)
= (2b + 3c)(4a + 3)

18. 10xy - 15xz + 6y - 9z
Group: (10xy - 15xz) + (6y - 9z)
Factor first group: 5x(2y - 3z)
Factor second group: 3(2y - 3z)
= (2y - 3z)(5x + 3)

19. 14mn + 21mp + 6n + 9p
Group: (14mn + 21mp) + (6n + 9p)
Factor first group: 7m(2n + 3p)
Factor second group: 3(2n + 3p)
= (2n + 3p)(7m + 3)

20. 12pq - 18pr - 8q + 12r
Group: (12pq - 18pr) + (-8q + 12r)
Factor first group: 6p(2q - 3r)
Factor second group: -4(2q - 3r)
= (2q - 3r)(6p - 4)
Factor out 2 from second bracket: = 2(2q - 3r)(3p - 2)

📌 Section C: 10 Advanced Grouping with Signs

Instructions: Factorise each expression. Pay careful attention to signs when grouping.

21. ax - bx + ay - by

22. 2x - 3y + 4xz - 6yz

23. 5a - 10b + 3ac - 6bc

24. 4p - 6q - 2pr + 3qr

25. 3m + 9n - 2mp - 6np

26. 8x - 12y - 6xz + 9yz

27. 10a + 15b - 4ac - 6bc

28. 6p - 9q + 8pr - 12qr

29. 14x + 21y - 6xz - 9yz

30. 12m - 18n - 8mp + 12np

✅ Solutions - Section C

21. ax - bx + ay - by
Group: (ax - bx) + (ay - by)
Factor first group: x(a - b)
Factor second group: y(a - b)
= (a - b)(x + y)

22. 2x - 3y + 4xz - 6yz
Group: (2x - 3y) + (4xz - 6yz)
Factor first group: (2x - 3y)
Factor second group: 2z(2x - 3y)
= (2x - 3y)(1 + 2z)

23. 5a - 10b + 3ac - 6bc
Group: (5a - 10b) + (3ac - 6bc)
Factor first group: 5(a - 2b)
Factor second group: 3c(a - 2b)
= (a - 2b)(5 + 3c)

24. 4p - 6q - 2pr + 3qr
Group: (4p - 6q) + (-2pr + 3qr)
Factor first group: 2(2p - 3q)
Factor second group: -r(2p - 3q)
= (2p - 3q)(2 - r) = (2p - 3q)(2 - r)

25. 3m + 9n - 2mp - 6np
Group: (3m + 9n) + (-2mp - 6np)
Factor first group: 3(m + 3n)
Factor second group: -2p(m + 3n)
= (m + 3n)(3 - 2p)

26. 8x - 12y - 6xz + 9yz
Group: (8x - 12y) + (-6xz + 9yz)
Factor first group: 4(2x - 3y)
Factor second group: -3z(2x - 3y)
= (2x - 3y)(4 - 3z) = (2x - 3y)(4 - 3z)

27. 10a + 15b - 4ac - 6bc
Group: (10a + 15b) + (-4ac - 6bc)
Factor first group: 5(2a + 3b)
Factor second group: -2c(2a + 3b)
= (2a + 3b)(5 - 2c)

28. 6p - 9q + 8pr - 12qr
Group: (6p - 9q) + (8pr - 12qr)
Factor first group: 3(2p - 3q)
Factor second group: 4r(2p - 3q)
= (2p - 3q)(3 + 4r)

29. 14x + 21y - 6xz - 9yz
Group: (14x + 21y) + (-6xz - 9yz)
Factor first group: 7(2x + 3y)
Factor second group: -3z(2x + 3y)
= (2x + 3y)(7 - 3z)

30. 12m - 18n - 8mp + 12np
Group: (12m - 18n) + (-8mp + 12np)
Factor first group: 6(2m - 3n)
Factor second group: -4p(2m - 3n)
= (2m - 3n)(6 - 4p)
Factor out 2 from second bracket: = 2(2m - 3n)(3 - 2p)

📊 Quick Reference Summary

Section	Type	Example	Factorised Form
A	Basic grouping	ax + bx + ay + by	(a + b)(x + y)
A	Number coefficients	2a + 3b + 4a + 6b	3(2a + 3b)
B	Mixed letters	2ax + 3bx + 2ay + 3by	(2a + 3b)(x + y)
B	With numbers	8ab + 12ac + 6b + 9c	(2b + 3c)(4a + 3)
C	With signs	ax - bx + ay - by	(a - b)(x + y)
C	Mixed signs	4p - 6q - 2pr + 3qr	(2p - 3q)(2 - r)

📝 Key Takeaways for Factorising by Grouping:

Always look for common factors in each group first
The two groups should have a common bracket after factoring
Be careful with signs - when factoring out a negative, the signs in the bracket may change
Sometimes you need to rearrange terms before grouping
Always check your answer by expanding the brackets to verify you get back the original expression
Factor out any common numerical factors from the final expression if possible

End of Factorising Linear Expressions - Grouping Pairs

Quadratic Equations - PSF Approach (a = 1)

QUADRATIC EQUATIONS - PSF Approach (a = 1)

📘 PSF Approach for a = 1

For quadratic in form x² + bx + c = 0 where a = 1:

P = Product = c
S = Sum = b
F = Find two numbers whose product = P and sum = S
Write as (x + m)(x + n) where m and n are the two numbers
Solve for x (if equation equals 0 or constant)

Example 1 (Positive): x² + 7x + 12 = 0

P = 12, S = 7
Numbers: 3 and 4 (3 × 4 = 12, 3 + 4 = 7)
(x + 3)(x + 4) = 0
x = -3 or x = -4

Example 2 (Negative): x² - 5x + 6 = 0

P = 6, S = -5
Numbers: -2 and -3 (-2 × -3 = 6, -2 + -3 = -5)
(x - 2)(x - 3) = 0
x = 2 or x = 3

Example 3 (Mixed signs): x² + 2x - 15 = 0

P = -15, S = 2
Numbers: 5 and -3 (5 × -3 = -15, 5 + (-3) = 2)
(x + 5)(x - 3) = 0
x = -5 or x = 3

⚠️ Important: For expressions (no equals sign), we only factorise. For equations with = 0 or = c, we solve for x. When an equation equals a constant c, first rearrange to = 0 by moving all terms to one side.

🔍 Sign Rules: When factorising (x + m)(x + n):

If c is positive, m and n have the same sign (both + or both -)
If c is negative, m and n have opposite signs
The sum b determines the actual signs

📌 Section A: 10 Expressions (Factorise Only)

Instructions: Factorise each quadratic expression completely. Find two numbers whose product = c and sum = b, then write as (x + m)(x + n).

These are expressions, not equations - so no solving for x required.

1. x² + 8x + 15

2. x² + 9x + 20

3. x² - 7x + 12

4. x² - 8x + 15

5. x² + 4x - 12

6. x² - 2x - 15

7. x² + 10x + 21

8. x² - 11x + 28

9. x² + 3x - 18

10. x² - 4x - 21

✅ Solutions - Section A

1. x² + 8x + 15 = (x + 3)(x + 5)

2. x² + 9x + 20 = (x + 4)(x + 5)

3. x² - 7x + 12 = (x - 3)(x - 4)

4. x² - 8x + 15 = (x - 3)(x - 5)

5. x² + 4x - 12 = (x + 6)(x - 2)

6. x² - 2x - 15 = (x - 5)(x + 3)

7. x² + 10x + 21 = (x + 3)(x + 7)

8. x² - 11x + 28 = (x - 4)(x - 7)

9. x² + 3x - 18 = (x + 6)(x - 3)

10. x² - 4x - 21 = (x - 7)(x + 3)

📌 Section B: 5 Equations in Form = 0

Instructions: Solve each quadratic equation. Find two numbers whose product = c and sum = b, then set each factor equal to zero.

These equations are already set equal to zero.

11. x² + 5x + 6 = 0

12. x² - 9x + 20 = 0

13. x² + x - 12 = 0

14. x² - 4x - 12 = 0

15. x² - 8x + 16 = 0

✅ Solutions - Section B

11. x² + 5x + 6 = 0
P = 6, S = 5
Numbers: 2 and 3 (2 × 3 = 6, 2 + 3 = 5)
(x + 2)(x + 3) = 0
x = -2 or x = -3

12. x² - 9x + 20 = 0
P = 20, S = -9
Numbers: -4 and -5 (-4 × -5 = 20, -4 + -5 = -9)
(x - 4)(x - 5) = 0
x = 4 or x = 5

13. x² + x - 12 = 0
P = -12, S = 1
Numbers: 4 and -3 (4 × -3 = -12, 4 + (-3) = 1)
(x + 4)(x - 3) = 0
x = -4 or x = 3

14. x² - 4x - 12 = 0
P = -12, S = -4
Numbers: -6 and 2 (-6 × 2 = -12, -6 + 2 = -4)
(x - 6)(x + 2) = 0
x = 6 or x = -2

15. x² - 8x + 16 = 0
P = 16, S = -8
Numbers: -4 and -4 (-4 × -4 = 16, -4 + -4 = -8)
(x - 4)(x - 4) = 0
x = 4 (repeated root / double root)

📌 Section C: 5 Equations in Form = c

Instructions: Solve each quadratic equation. First, rearrange to the form = 0 by moving the constant term to the left side. Then factorise and solve.

Remember: Whatever you do to one side, you must do to the other!

16. x² + 6x + 5 = 12

17. x² - 7x + 3 = 11

18. x² + 4x - 2 = 10

19. x² - 5x - 1 = 13

20. x² + 8x + 7 = 15

✅ Solutions - Section C

16. x² + 6x + 5 = 12
Step 1: Bring all terms to one side
x² + 6x + 5 - 12 = 0
x² + 6x - 7 = 0
P = -7, S = 6
Numbers: 7 and -1 (7 × -1 = -7, 7 + (-1) = 6)
(x + 7)(x - 1) = 0
x = -7 or x = 1

17. x² - 7x + 3 = 11
Step 1: Bring all terms to one side
x² - 7x + 3 - 11 = 0
x² - 7x - 8 = 0
P = -8, S = -7
Numbers: -8 and 1 (-8 × 1 = -8, -8 + 1 = -7)
(x - 8)(x + 1) = 0
x = 8 or x = -1

18. x² + 4x - 2 = 10
Step 1: Bring all terms to one side
x² + 4x - 2 - 10 = 0
x² + 4x - 12 = 0
P = -12, S = 4
Numbers: 6 and -2 (6 × -2 = -12, 6 + (-2) = 4)
(x + 6)(x - 2) = 0
x = -6 or x = 2

19. x² - 5x - 1 = 13
Step 1: Bring all terms to one side
x² - 5x - 1 - 13 = 0
x² - 5x - 14 = 0
P = -14, S = -5
Numbers: -7 and 2 (-7 × 2 = -14, -7 + 2 = -5)
(x - 7)(x + 2) = 0
x = 7 or x = -2

20. x² + 8x + 7 = 15
Step 1: Bring all terms to one side
x² + 8x + 7 - 15 = 0
x² + 8x - 8 = 0
P = -8, S = 8
Numbers: ? Let's find factors of -8 with sum 8
9 and -1 → product = -9 ❌
8 and -1 → product = -8, sum = 7 ❌ (need 8)
4 and -2 → product = -8, sum = 2 ❌
8 and -1. Something's not right...
Actually: 8 × (-1) = -8, 8 + (-1) = 7 ❌ (need 8)
10 and -2 = 8? No, product = -20 ❌

This quadratic does not factor nicely with integers.
Using quadratic formula: x = [-8 ± √(8² - 4×1×(-8))] / 2
x = [-8 ± √(64 + 32)] / 2
x = [-8 ± √96] / 2
x = [-8 ± 9.80] / 2
x = (-8 + 9.80)/2 = 1.80/2 = 0.90
x = (-8 - 9.80)/2 = -17.80/2 = -8.90

📊 Quick Reference Summary Table

#	Type	Problem	Solutions / Factors
1	Expression	x² + 8x + 15	(x + 3)(x + 5)
2	Expression	x² + 9x + 20	(x + 4)(x + 5)
3	Expression	x² - 7x + 12	(x - 3)(x - 4)
4	Expression	x² - 8x + 15	(x - 3)(x - 5)
5	Expression	x² + 4x - 12	(x + 6)(x - 2)
6	Expression	x² - 2x - 15	(x - 5)(x + 3)
7	Expression	x² + 10x + 21	(x + 3)(x + 7)
8	Expression	x² - 11x + 28	(x - 4)(x - 7)
9	Expression	x² + 3x - 18	(x + 6)(x - 3)
10	Expression	x² - 4x - 21	(x - 7)(x + 3)
11	= 0	x² + 5x + 6 = 0	x = -2, -3
12	= 0	x² - 9x + 20 = 0	x = 4, 5
13	= 0	x² + x - 12 = 0	x = -4, 3
14	= 0	x² - 4x - 12 = 0	x = 6, -2
15	= 0	x² - 8x + 16 = 0	x = 4 (double root)
16	= 12	x² + 6x + 5 = 12	x = -7, 1
17	= 11	x² - 7x + 3 = 11	x = 8, -1
18	= 10	x² + 4x - 2 = 10	x = -6, 2
19	= 13	x² - 5x - 1 = 13	x = 7, -2
20	= 15	x² + 8x + 7 = 15	x ≈ 0.90, -8.90

📝 Key Takeaways for a = 1:

Expressions → Factorise only as (x + m)(x + n)
= 0 equations → Factorise and set each factor = 0
= c equations → Rearrange to = 0 first, then solve
Look for two numbers that multiply to c and add to b
If c is positive, both numbers have the same sign (both + or both -)
If c is negative, the numbers have opposite signs
The double root (perfect square trinomial) occurs when the two numbers are equal

End of PSF Approach Practice Problems (a = 1)

Quadratic Equations - PSF Approach (a > 1)

QUADRATIC EQUATIONS - PSF Approach (a > 1)

📘 PSF Approach Recap

For quadratic in form ax² + bx + c = 0 where a > 1:

P = Product = a × c
S = Sum = b
F = Find two numbers whose product = P and sum = S
Split the middle term using these two numbers
Factor by grouping (pairing terms)
Solve for x (if equation equals 0 or constant)

Example: 2x² + 7x + 3 = 0

P = 2 × 3 = 6
S = 7
Numbers: 1 and 6 (1 × 6 = 6, 1 + 6 = 7)
2x² + 1x + 6x + 3 = 0
x(2x + 1) + 3(2x + 1) = 0
(2x + 1)(x + 3) = 0
x = -½ or x = -3

📌 Section A: 10 Expressions (Factorise Only)

Instructions: Factorise each quadratic expression completely. Look for two numbers whose product = a × c and sum = b, then split the middle term and factor by grouping.

These are expressions, not equations - so no solving for x required.

1. 2x² + 9x + 4

2. 3x² + 11x + 6

3. 4x² + 13x + 3

4. 5x² + 14x + 8

5. 2x² - 7x + 3

6. 3x² - 13x + 4

7. 4x² - 17x + 15

8. 6x² + 23x + 20

9. 5x² - 21x + 18

10. 7x² + 24x + 9

✅ Solutions - Section A

1. 2x² + 9x + 4 = (2x + 1)(x + 4)

2. 3x² + 11x + 6 = (3x + 2)(x + 3)

3. 4x² + 13x + 3 = (4x + 1)(x + 3)

4. 5x² + 14x + 8 = (5x + 4)(x + 2)

5. 2x² - 7x + 3 = (2x - 1)(x - 3)

6. 3x² - 13x + 4 = (3x - 1)(x - 4)

7. 4x² - 17x + 15 = (4x - 5)(x - 3)

8. 6x² + 23x + 20 = (3x + 4)(2x + 5)

9. 5x² - 21x + 18 = (5x - 6)(x - 3)

10. 7x² + 24x + 9 = (7x + 3)(x + 3)

📌 Section B: 5 Equations in Form = 0

Instructions: Solve each quadratic equation. Use the PSF approach to factorise, then set each factor equal to zero to find the solutions.

These equations are already set equal to zero.

11. 2x² + 9x - 5 = 0

12. 3x² - 10x - 8 = 0

13. 4x² + 5x - 6 = 0

14. 5x² - 12x - 9 = 0

15. 6x² + 7x - 10 = 0

✅ Solutions - Section B

11. 2x² + 9x - 5 = 0
P = 2 × (-5) = -10, S = 9
Numbers: 10 and -1 (10 × -1 = -10, 10 + (-1) = 9)
2x² + 10x - x - 5 = 0
2x(x + 5) - 1(x + 5) = 0
(x + 5)(2x - 1) = 0
x = -5 or x = ½

12. 3x² - 10x - 8 = 0
P = 3 × (-8) = -24, S = -10
Numbers: -12 and 2 (-12 × 2 = -24, -12 + 2 = -10)
3x² - 12x + 2x - 8 = 0
3x(x - 4) + 2(x - 4) = 0
(x - 4)(3x + 2) = 0
x = 4 or x = -²⁄₃

13. 4x² + 5x - 6 = 0
P = 4 × (-6) = -24, S = 5
Numbers: 8 and -3 (8 × -3 = -24, 8 + (-3) = 5)
4x² + 8x - 3x - 6 = 0
4x(x + 2) - 3(x + 2) = 0
(x + 2)(4x - 3) = 0
x = -2 or x = ¾

14. 5x² - 12x - 9 = 0
P = 5 × (-9) = -45, S = -12
Numbers: -15 and 3 (-15 × 3 = -45, -15 + 3 = -12)
5x² - 15x + 3x - 9 = 0
5x(x - 3) + 3(x - 3) = 0
(x - 3)(5x + 3) = 0
x = 3 or x = -³⁄₅

15. 6x² + 7x - 10 = 0
P = 6 × (-10) = -60, S = 7
Numbers: 12 and -5 (12 × -5 = -60, 12 + (-5) = 7)
6x² + 12x - 5x - 10 = 0
6x(x + 2) - 5(x + 2) = 0
(x + 2)(6x - 5) = 0
x = -2 or x = ⁵⁄₆

📌 Section C: 5 Equations in Form = c

Instructions: Solve each quadratic equation. First, rearrange to the form = 0 by moving the constant term to the left side. Then apply the PSF approach to factorise and solve.

Remember: Whatever you do to one side, you must do to the other!

16. 2x² + 8x - 3 = 7

17. 3x² - 5x + 2 = 14

18. 4x² + 6x - 1 = 11

19. 5x² - 9x + 4 = 16

20. 6x² + 11x - 2 = 23

✅ Solutions - Section C

16. 2x² + 8x - 3 = 7
Step 1: Bring all terms to one side
2x² + 8x - 3 - 7 = 0
2x² + 8x - 10 = 0
Step 2: Divide by 2 (optional)
x² + 4x - 5 = 0
P = 1 × (-5) = -5, S = 4
Numbers: 5 and -1 (5 × -1 = -5, 5 + (-1) = 4)
x² + 5x - x - 5 = 0
x(x + 5) - 1(x + 5) = 0
(x + 5)(x - 1) = 0
x = -5 or x = 1

17. 3x² - 5x + 2 = 14
Step 1: Bring all terms to one side
3x² - 5x + 2 - 14 = 0
3x² - 5x - 12 = 0
P = 3 × (-12) = -36, S = -5
Numbers: -9 and 4 (-9 × 4 = -36, -9 + 4 = -5)
3x² - 9x + 4x - 12 = 0
3x(x - 3) + 4(x - 3) = 0
(x - 3)(3x + 4) = 0
x = 3 or x = -⁴⁄₃

18. 4x² + 6x - 1 = 11
Step 1: Bring all terms to one side
4x² + 6x - 1 - 11 = 0
4x² + 6x - 12 = 0
Step 2: Divide by 2 (optional)
2x² + 3x - 6 = 0
P = 2 × (-6) = -12, S = 3
Numbers: 6 and -3 (6 × -3 = -12, 6 + (-3) = 3)
2x² + 6x - 3x - 6 = 0
2x(x + 3) - 3(x + 3) = 0
(x + 3)(2x - 3) = 0
x = -3 or x = ³⁄₂

19. 5x² - 9x + 4 = 16
Step 1: Bring all terms to one side
5x² - 9x + 4 - 16 = 0
5x² - 9x - 12 = 0
P = 5 × (-12) = -60, S = -9
Numbers: -15 and 4 (-15 × 4 = -60, -15 + 4 = -9)
5x² - 15x + 4x - 12 = 0
5x(x - 3) + 4(x - 3) = 0
(x - 3)(5x + 4) = 0
x = 3 or x = -⁴⁄₅

20. 6x² + 11x - 2 = 23
Step 1: Bring all terms to one side
6x² + 11x - 2 - 23 = 0
6x² + 11x - 25 = 0
P = 6 × (-25) = -150, S = 11
Numbers: 25 and -6 → sum = 19 ❌ (need 11)
30 and -5 → sum = 25 ❌
15 and -10 → sum = 5 ❌
This quadratic does not factor nicely with integers.
Using quadratic formula: x = [-11 ± √(11² - 4×6×(-25))] / (2×6)
x = [-11 ± √(121 + 600)] / 12
x = [-11 ± √721] / 12
x = [-11 ± 26.85] / 12
x = (-11 + 26.85)/12 = 15.85/12 = 1.32
x = (-11 - 26.85)/12 = -37.85/12 = -3.15

📊 Quick Reference Summary Table

#	Type	Problem	Solutions / Factors
1	Expression	2x² + 9x + 4	(2x + 1)(x + 4)
2	Expression	3x² + 11x + 6	(3x + 2)(x + 3)
3	Expression	4x² + 13x + 3	(4x + 1)(x + 3)
4	Expression	5x² + 14x + 8	(5x + 4)(x + 2)
5	Expression	2x² - 7x + 3	(2x - 1)(x - 3)
6	Expression	3x² - 13x + 4	(3x - 1)(x - 4)
7	Expression	4x² - 17x + 15	(4x - 5)(x - 3)
8	Expression	6x² + 23x + 20	(3x + 4)(2x + 5)
9	Expression	5x² - 21x + 18	(5x - 6)(x - 3)
10	Expression	7x² + 24x + 9	(7x + 3)(x + 3)
11	= 0	2x² + 9x - 5 = 0	x = -5, ½
12	= 0	3x² - 10x - 8 = 0	x = 4, -²⁄₃
13	= 0	4x² + 5x - 6 = 0	x = -2, ¾
14	= 0	5x² - 12x - 9 = 0	x = 3, -³⁄₅
15	= 0	6x² + 7x - 10 = 0	x = -2, ⁵⁄₆
16	= 7	2x² + 8x - 3 = 7	x = -5, 1
17	= 14	3x² - 5x + 2 = 14	x = 3, -⁴⁄₃
18	= 11	4x² + 6x - 1 = 11	x = -3, ³⁄₂
19	= 16	5x² - 9x + 4 = 16	x = 3, -⁴⁄₅
20	= 23	6x² + 11x - 2 = 23	x ≈ 1.32, -3.15

📝 Key Takeaways:

Expressions → Factorise only
= 0 equations → Factorise and solve
= c equations → Rearrange to = 0 first, then solve
Always find P = a × c and S = b first
Check your factors by expanding back

End of PSF Approach Practice Problems

Quadratic Equations - PSF Approach (a < 0)

QUADRATIC EQUATIONS - PSF Approach (a < 0)

📘 PSF Approach for Negative Leading Coefficient (a < 0)

For quadratic in form ax² + bx + c = 0 where a < 0:

Method 1: Factor out -1 first

Factor out -1: -(x² - bx - c) = 0 or -ax² - bx - c = 0 becomes ax² + bx + c = 0 after multiplying by -1
Multiply both sides by -1 to make a > 0
Apply PSF approach as normal with positive a
Remember that multiplying by -1 does not change the solutions

Method 2: Direct PSF with negative a

P = Product = a × c (this will be negative if c is positive)
S = Sum = b
F = Find two numbers whose product = P and sum = S
Split the middle term using these two numbers
Factor by grouping - careful with signs!

Example 1 (Factor out -1): -x² + 5x - 6 = 0

Multiply by -1: x² - 5x + 6 = 0
P = 6, S = -5
Numbers: -2 and -3 (-2 × -3 = 6, -2 + -3 = -5)
(x - 2)(x - 3) = 0
x = 2 or x = 3

Example 2 (Direct): -2x² + 7x - 3 = 0

P = (-2) × (-3) = 6, S = 7
Numbers: 1 and 6 (1 × 6 = 6, 1 + 6 = 7)
-2x² + 1x + 6x - 3 = 0
-x(2x - 1) + 3(2x - 1) = 0? Let's check carefully
Better to factor out -1 first: -(2x² - 7x + 3) = 0
2x² - 7x + 3 = 0
P = 2 × 3 = 6, S = -7
Numbers: -1 and -6 (-1 × -6 = 6, -1 + -6 = -7)
2x² - 1x - 6x + 3 = 0
x(2x - 1) - 3(2x - 1) = 0
(2x - 1)(x - 3) = 0
x = ½ or x = 3

⚠️ Important Warning: When a is negative, the parabola opens downward. The PSF approach still works, but you must be extremely careful with signs. The safest approach is to factor out -1 first and work with a positive leading coefficient.

🔍 Remember: Multiplying or dividing an equation by -1 does NOT change the solutions!

📌 Section A: 10 Expressions (Factorise Only)

Instructions: Factorise each quadratic expression completely. For expressions with a negative leading coefficient, you may either factor out -1 first or factor directly. Present your answer in factorised form.

These are expressions, not equations - so no solving for x required.

1. -x² + 5x - 6

2. -x² + 7x - 10

3. -x² + 4x + 5

4. -x² + 2x + 8

5. -2x² + 7x - 3

6. -3x² + 10x - 8

7. -4x² + 12x - 5

8. -2x² + 9x - 4

9. -3x² + 11x - 6

10. -5x² + 16x - 12

✅ Solutions - Section A

1. -x² + 5x - 6
Factor out -1: -(x² - 5x + 6)
x² - 5x + 6 = (x - 2)(x - 3)
Therefore: -(x - 2)(x - 3) or (-x + 2)(x - 3)

2. -x² + 7x - 10
Factor out -1: -(x² - 7x + 10)
x² - 7x + 10 = (x - 2)(x - 5)
Therefore: -(x - 2)(x - 5)

3. -x² + 4x + 5
Factor out -1: -(x² - 4x - 5)
x² - 4x - 5 = (x - 5)(x + 1)
Therefore: -(x - 5)(x + 1) or (-x + 5)(x + 1)

4. -x² + 2x + 8
Factor out -1: -(x² - 2x - 8)
x² - 2x - 8 = (x - 4)(x + 2)
Therefore: -(x - 4)(x + 2)

5. -2x² + 7x - 3
Factor out -1: -(2x² - 7x + 3)
2x² - 7x + 3: P = 6, S = -7 → numbers -1, -6
2x² - x - 6x + 3 = x(2x - 1) - 3(2x - 1) = (2x - 1)(x - 3)
Therefore: -(2x - 1)(x - 3)

6. -3x² + 10x - 8
Factor out -1: -(3x² - 10x + 8)
3x² - 10x + 8: P = 24, S = -10 → numbers -4, -6
3x² - 4x - 6x + 8 = x(3x - 4) - 2(3x - 4) = (3x - 4)(x - 2)
Therefore: -(3x - 4)(x - 2)

7. -4x² + 12x - 5
Factor out -1: -(4x² - 12x + 5)
4x² - 12x + 5: P = 20, S = -12 → numbers -2, -10
4x² - 2x - 10x + 5 = 2x(2x - 1) - 5(2x - 1) = (2x - 1)(2x - 5)
Therefore: -(2x - 1)(2x - 5)

8. -2x² + 9x - 4
Factor out -1: -(2x² - 9x + 4)
2x² - 9x + 4: P = 8, S = -9 → numbers -1, -8
2x² - x - 8x + 4 = x(2x - 1) - 4(2x - 1) = (2x - 1)(x - 4)
Therefore: -(2x - 1)(x - 4)

9. -3x² + 11x - 6
Factor out -1: -(3x² - 11x + 6)
3x² - 11x + 6: P = 18, S = -11 → numbers -2, -9
3x² - 2x - 9x + 6 = x(3x - 2) - 3(3x - 2) = (3x - 2)(x - 3)
Therefore: -(3x - 2)(x - 3)

10. -5x² + 16x - 12
Factor out -1: -(5x² - 16x + 12)
5x² - 16x + 12: P = 60, S = -16 → numbers -6, -10
5x² - 6x - 10x + 12 = x(5x - 6) - 2(5x - 6) = (5x - 6)(x - 2)
Therefore: -(5x - 6)(x - 2)

📌 Section B: 5 Equations in Form = 0

Instructions: Solve each quadratic equation. Remember that you can multiply both sides by -1 to make the leading coefficient positive before applying PSF.

These equations are already set equal to zero.

11. -x² + 8x - 15 = 0

12. -x² + 6x - 8 = 0

13. -2x² + 9x - 4 = 0

14. -3x² + 14x - 8 = 0

15. -4x² + 17x - 15 = 0

✅ Solutions - Section B

11. -x² + 8x - 15 = 0
Multiply by -1: x² - 8x + 15 = 0
P = 15, S = -8
Numbers: -3 and -5 (-3 × -5 = 15, -3 + -5 = -8)
(x - 3)(x - 5) = 0
x = 3 or x = 5

12. -x² + 6x - 8 = 0
Multiply by -1: x² - 6x + 8 = 0
P = 8, S = -6
Numbers: -2 and -4 (-2 × -4 = 8, -2 + -4 = -6)
(x - 2)(x - 4) = 0
x = 2 or x = 4

13. -2x² + 9x - 4 = 0
Multiply by -1: 2x² - 9x + 4 = 0
P = 2 × 4 = 8, S = -9
Numbers: -1 and -8 (-1 × -8 = 8, -1 + -8 = -9)
2x² - x - 8x + 4 = 0
x(2x - 1) - 4(2x - 1) = 0
(2x - 1)(x - 4) = 0
x = ½ or x = 4

14. -3x² + 14x - 8 = 0
Multiply by -1: 3x² - 14x + 8 = 0
P = 3 × 8 = 24, S = -14
Numbers: -2 and -12 (-2 × -12 = 24, -2 + -12 = -14)
3x² - 2x - 12x + 8 = 0
x(3x - 2) - 4(3x - 2) = 0
(3x - 2)(x - 4) = 0
x = ²⁄₃ or x = 4

15. -4x² + 17x - 15 = 0
Multiply by -1: 4x² - 17x + 15 = 0
P = 4 × 15 = 60, S = -17
Numbers: -5 and -12 (-5 × -12 = 60, -5 + -12 = -17)
4x² - 5x - 12x + 15 = 0
x(4x - 5) - 3(4x - 5) = 0
(4x - 5)(x - 3) = 0
x = ⁵⁄₄ or x = 3

📌 Section C: 5 Equations in Form = c

Instructions: Solve each quadratic equation. First, rearrange to the form = 0. Then multiply by -1 if needed to make the leading coefficient positive before applying PSF.

Remember: Move all terms to one side first!

16. -x² + 7x - 5 = 7

17. -x² + 9x + 2 = 12

18. -2x² + 8x - 3 = 5

19. -3x² + 10x + 1 = 13

20. -4x² + 15x - 2 = 10

✅ Solutions - Section C

16. -x² + 7x - 5 = 7
Step 1: Bring all terms to one side
-x² + 7x - 5 - 7 = 0
-x² + 7x - 12 = 0
Step 2: Multiply by -1
x² - 7x + 12 = 0
P = 12, S = -7
Numbers: -3 and -4 (-3 × -4 = 12, -3 + -4 = -7)
(x - 3)(x - 4) = 0
x = 3 or x = 4

17. -x² + 9x + 2 = 12
Step 1: Bring all terms to one side
-x² + 9x + 2 - 12 = 0
-x² + 9x - 10 = 0
Step 2: Multiply by -1
x² - 9x + 10 = 0
P = 10, S = -9
Numbers: ? Factors of 10 with sum -9: -4 and -5? -4 × -5 = 20 ❌
-2 and -5? -2 × -5 = 10, -2 + -5 = -7 ❌
-1 and -10? -1 × -10 = 10, -1 + -10 = -11 ❌
This quadratic may not factor nicely with integers.
Using quadratic formula: x = [9 ± √(81 - 40)] / 2 = [9 ± √41] / 2
x = (9 + 6.40)/2 = 15.40/2 = 7.70
x = (9 - 6.40)/2 = 2.60/2 = 1.30

18. -2x² + 8x - 3 = 5
Step 1: Bring all terms to one side
-2x² + 8x - 3 - 5 = 0
-2x² + 8x - 8 = 0
Step 2: Multiply by -1
2x² - 8x + 8 = 0
Step 3: Divide by 2
x² - 4x + 4 = 0
(x - 2)(x - 2) = 0
x = 2 (double root)

19. -3x² + 10x + 1 = 13
Step 1: Bring all terms to one side
-3x² + 10x + 1 - 13 = 0
-3x² + 10x - 12 = 0
Step 2: Multiply by -1
3x² - 10x + 12 = 0
P = 3 × 12 = 36, S = -10
Numbers: -4 and -6? -4 × -6 = 24 ❌
-3 and -12? -3 × -12 = 36, -3 + -12 = -15 ❌
-2 and -18? -2 × -18 = 36, -2 + -18 = -20 ❌
This quadratic may not factor nicely with integers.
Using quadratic formula: x = [10 ± √(100 - 144)] / 6 = [10 ± √(-44)] / 6
Discriminant is negative! No real solutions (complex roots)

20. -4x² + 15x - 2 = 10
Step 1: Bring all terms to one side
-4x² + 15x - 2 - 10 = 0
-4x² + 15x - 12 = 0
Step 2: Multiply by -1
4x² - 15x + 12 = 0
P = 4 × 12 = 48, S = -15
Numbers: -3 and -12? -3 × -12 = 36 ❌
-4 and -12? -4 × -12 = 48, -4 + -12 = -16 ❌
-6 and -8? -6 × -8 = 48, -6 + -8 = -14 ❌
This quadratic may not factor nicely with integers.
Using quadratic formula: x = [15 ± √(225 - 192)] / 8 = [15 ± √33] / 8
x = (15 + 5.74)/8 = 20.74/8 = 2.59
x = (15 - 5.74)/8 = 9.26/8 = 1.16

📊 Quick Reference Summary Table

#	Type	Problem	Solutions / Factors
1	Expression	-x² + 5x - 6	-(x - 2)(x - 3)
2	Expression	-x² + 7x - 10	-(x - 2)(x - 5)
3	Expression	-x² + 4x + 5	-(x - 5)(x + 1)
4	Expression	-x² + 2x + 8	-(x - 4)(x + 2)
5	Expression	-2x² + 7x - 3	-(2x - 1)(x - 3)
6	Expression	-3x² + 10x - 8	-(3x - 4)(x - 2)
7	Expression	-4x² + 12x - 5	-(2x - 1)(2x - 5)
8	Expression	-2x² + 9x - 4	-(2x - 1)(x - 4)
9	Expression	-3x² + 11x - 6	-(3x - 2)(x - 3)
10	Expression	-5x² + 16x - 12	-(5x - 6)(x - 2)
11	= 0	-x² + 8x - 15 = 0	x = 3, 5
12	= 0	-x² + 6x - 8 = 0	x = 2, 4
13	= 0	-2x² + 9x - 4 = 0	x = ½, 4
14	= 0	-3x² + 14x - 8 = 0	x = ²⁄₃, 4
15	= 0	-4x² + 17x - 15 = 0	x = ⁵⁄₄, 3
16	= 7	-x² + 7x - 5 = 7	x = 3, 4
17	= 12	-x² + 9x + 2 = 12	x ≈ 7.70, 1.30
18	= 5	-2x² + 8x - 3 = 5	x = 2 (double root)
19	= 13	-3x² + 10x + 1 = 13	No real solutions
20	= 10	-4x² + 15x - 2 = 10	x ≈ 2.59, 1.16

📝 Key Takeaways for a < 0:

Always consider multiplying by -1 first to make a > 0 - it's safer and reduces sign errors
Multiplying by -1 does NOT change the solutions of an equation
For expressions, you can leave the answer as -(factorised form)
Watch for negative discriminants (no real solutions)
Some quadratics with a < 0 may not factor nicely - use the quadratic formula
The parabola opens downward when a < 0, meaning the vertex is a maximum point

End of PSF Approach Practice Problems (a < 0)