Solving Quadratic Equations – Grade 9 IGCSE Maths

Welcome to Quadratic Equations!

A quadratic equation has the form ax² + bx + c = 0, where the highest power is 2. Quadratics appear in physics (projectile motion), economics (profit maximisation), engineering, and geometry. You will master three solving methods — factorising, completing the square, and the quadratic formula — as well as using the discriminant to predict the nature of solutions.

x = (−b ± √(b² − 4ac)) / 2a Discriminant: Δ = b² − 4ac

Factorising Method

Fastest when it works

Completing the Square

Write as (x+p)² + q

Quadratic Formula

Works for any quadratic

The Discriminant

Δ = b² − 4ac predicts roots

Nature of Roots

0, 1 or 2 real solutions

Interactive Solver

Graph and solve any quadratic

1. Solving by Factorising

Factorising is the quickest method when the quadratic factorises neatly. Rearrange to ax² + bx + c = 0, factorise the left side, then apply the zero product property.

If (x + p)(x + q) = 0, then x = −p or x = −q

Solve x² + 5x + 6 = 0
Factorise: need two numbers with product 6, sum 5 → 2 and 3
(x + 2)(x + 3) = 0
x = −2 or x = −3

Solve x² − x − 12 = 0
Need product −12, sum −1 → −4 and 3
(x − 4)(x + 3) = 0
x = 4 or x = −3

Always rearrange to = 0 first. If the equation is x² = 5x, write it as x² − 5x = 0, factorise as x(x − 5) = 0, giving x = 0 or x = 5.

2. Completing the Square

Completing the square rewrites any quadratic in the form (x + p)² + q. This form reveals the vertex of the parabola and allows exact solving without the formula.

x² + bx + c = (x + b/2)² − (b/2)² + c so p = b/2, q = c − (b/2)²

Complete the square for x² + 6x + 2:
Half of 6 = 3 → (x + 3)² = x² + 6x + 9
x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7
q = −7

Solve x² + 6x + 2 = 0 using completed square:
(x + 3)² = 7
x + 3 = ±√7
x = −3 ± √7 ≈ −0.354 or −5.646

3. The Quadratic Formula

Derived by completing the square on the general form, the quadratic formula works for every quadratic — even those that cannot be factorised over integers.

x = (−b ± √(b² − 4ac)) / (2a)

Solve 2x² + 3x − 5 = 0:
a = 2, b = 3, c = −5
Δ = 9 + 40 = 49
x = (−3 ± 7) / 4
x = 4/4 = 1 or x = −10/4 = −2.5
x = 1 or x = −2.5

Method selection: try factorising first (quick). If no obvious factors, use the quadratic formula or completing the square. The formula always works.

4. The Discriminant

The discriminant is the expression b² − 4ac inside the square root in the quadratic formula. It tells you how many real solutions exist without fully solving.

Δ = b² − 4ac

Δ > 0: Two distinct real roots (parabola crosses x-axis at two points)
Δ = 0: One repeated real root (parabola touches x-axis at one point)
Δ < 0: No real roots (parabola does not meet x-axis)

x² − 5x + 4: Δ = 25 − 16 = 9 → two roots (positive)
x² − 4x + 4: Δ = 16 − 16 = 0 → one repeated root
x² + 2x + 5: Δ = 4 − 20 = −16 → no real roots

Exam tip: you can be asked to show a quadratic has two equal roots — this means showing Δ = 0. Or to show no real roots — show Δ < 0.

5. Word Problems Leading to Quadratics

Example: A rectangle has width x cm and length (x + 3) cm. Area = 28 cm². Find x.
x(x + 3) = 28
x² + 3x − 28 = 0
(x + 7)(x − 4) = 0
x = 4 (since x must be positive)

In word problems: form the equation, solve both roots, then reject any root that doesn't make sense in context (e.g. negative lengths).

Example 1 — Factorise: x² + 7x + 12 = 0

Need: p × q = 12, p + q = 7. Try 3 and 4: 3 × 4 = 12, 3 + 4 = 7 ✓

(x + 3)(x + 4) = 0 → x = −3 or x = −4

Example 2 — Factorise: x² − x − 6 = 0

Need: p × q = −6, p + q = −1. Try −3 and 2: −3 × 2 = −6 ✓, −3 + 2 = −1 ✓

(x − 3)(x + 2) = 0 → x = 3 or x = −2

Example 3 — Quadratic Formula: x² − 5x + 3 = 0

a=1, b=−5, c=3. Δ = 25 − 12 = 13

x = (5 ± √13)/2. √13 ≈ 3.606

x₁ ≈ (5 + 3.606)/2 ≈ 4.30 x₂ ≈ (5 − 3.606)/2 ≈ 0.70

Example 4 — Discriminant of x² + 4x + 4

Δ = 4² − 4(1)(4) = 16 − 16 = 0 → one repeated root at x = −2

Example 5 — Complete the Square: x² + 6x

p = 6/2 = 3. (x + 3)² = x² + 6x + 9, so x² + 6x = (x + 3)² − 9

q = −9 (constant term in (x+3)² + q form)

Example 6 — Negative Discriminant: x² + 2x + 5

Δ = 4 − 20 = −16 < 0. No real solutions exist.

Example 7 — Word Problem

A number plus its square equals 20. Find the positive number.

x + x² = 20 → x² + x − 20 = 0 → (x + 5)(x − 4) = 0

x = 4 (positive) or x = −5 (rejected). Answer: 4

Quadratic Solver and Graph

Enter a, b, c for ax² + bx + c = 0. The tool shows the discriminant, all solution steps, and plots the parabola with roots marked.

a = b = c =

Enter coefficients and click Solve.

Exercise 1 — Solving by Factorising (Positive Root)

Solve each quadratic by factorising. Give the positive root (or the root with smaller absolute value if both are negative).

1. x² + 7x + 12 = 0. Positive root?

2. x² + 8x + 15 = 0. Roots are negative. Enter the one closest to zero (smaller magnitude).

3. x² + 9x + 20 = 0. Roots are negative. Smaller magnitude root?

4. x² − x − 6 = 0. Positive root?

5. x² + 3x − 28 = 0. Positive root?

6. x² + x − 30 = 0. Positive root?

7. x² + 2x − 3 = 0. Positive root?

8. x² − 2x − 48 = 0. Positive root?

9. x² − 2x − 63 = 0. Positive root?

10. x² − 2x − 80 = 0. Positive root?

Exercise 2 — Solving by Factorising (Negative Root Magnitude)

Solve by factorising. Give the magnitude (absolute value) of the negative root.

1. x² − x − 6 = 0. |Negative root|?

2. x² − 2x − 8 = 0. |Negative root|?

3. x² − x − 2 = 0. |Negative root|?

4. x² − x − 12 = 0. |Negative root|?

5. x² − 3x − 10 = 0. |Negative root|?

6. x² − 5x − 14 = 0. |Negative root|?

7. x² − 4x − 12 = 0. |Negative root|?

8. x² − 4x − 45 = 0. |Negative root|?

9. x² − 5x − 24 = 0. |Negative root|?

10. x² − 5x − 66 = 0. |Negative root|?

Exercise 3 — Quadratic Formula (Larger Root)

Use the quadratic formula to solve each equation. Give the larger root to 2 d.p.

1. x² − 5x + 3 = 0. Larger root to 2 d.p.?

2. x² − 7x + 3 = 0. Larger root to 2 d.p.?

3. x² − 4x + 1 = 0. Larger root to 2 d.p.?

4. x² − 9x + 5 = 0. Larger root to 2 d.p.?

5. x² − 5x + 2 = 0. Larger root to 2 d.p.?

6. x² − 10x + 8 = 0. Larger root to 2 d.p.?

7. x² − 3x + 1 = 0. Larger root to 2 d.p.?

8. x² − 6x − 2 = 0. Larger root to 2 d.p.?

9. x² − 5x − 1 = 0. Larger root to 2 d.p.?

10. x² − 11x + 5 = 0. Larger root to 2 d.p.?

Exercise 4 — The Discriminant

Calculate Δ = b² − 4ac for each quadratic. Give the exact integer value.

1. x² − 5x + 0 = 0 (a=1, b=−5, c=0). Δ?

2. x² − 7x + 0 = 0. Δ?

3. x² − 4x + 4 = 0. Δ?

4. x² − 3x + 2 = 0. Δ?

5. x² − 4x + 0 = 0. Δ?

6. x² + 2x + 3 = 0. Δ?

7. x² − 6x + 0 = 0. Δ?

8. x² − 2x + 0 = 0. Δ?

9. x² + 2x + 2 = 0. Δ?

10. x² − 10x + 0 = 0. Δ?

Exercise 5 — Completing the Square (Value of q)

Write each expression in the form (x + p)² + q. Give the value of q (the constant remaining after completing the square).

1. x² + 6x → (x + 3)² + q. Give q.

2. x² + 10x → (x + 5)² + q. Give q.

3. x² + 4x → (x + 2)² + q. Give q.

4. x² + 8x → (x + 4)² + q. Give q.

5. x² + 2x → (x + 1)² + q. Give q.

6. x² + 12x → (x + 6)² + q. Give q.

7. x² + 14x → (x + 7)² + q. Give q.

8. x² + 16x → (x + 8)² + q. Give q.

9. x² + 18x → (x + 9)² + q. Give q.

10. x² + 20x → (x + 10)² + q. Give q.

Practice — 20 Questions

Mixed quadratic practice. Q1–10: positive roots by factorising. Q11–20: magnitudes of negative roots by factorising.

1. x² + 7x + 12 = 0. Positive root?

2. x² + 8x + 15 = 0. Smaller |root|?

3. x² + 9x + 20 = 0. Smaller |root|?

4. x² − x − 6 = 0. Positive root?

5. x² + 3x − 28 = 0. Positive root?

6. x² + x − 30 = 0. Positive root?

7. x² + 2x − 3 = 0. Positive root?

8. x² − 2x − 48 = 0. Positive root?

9. x² − 2x − 63 = 0. Positive root?

10. x² − 2x − 80 = 0. Positive root?

11. x² − x − 6 = 0. |Negative root|?

12. x² − 2x − 8 = 0. |Negative root|?

13. x² − x − 2 = 0. |Negative root|?

14. x² − x − 12 = 0. |Negative root|?

15. x² − 3x − 10 = 0. |Negative root|?

16. x² − 5x − 14 = 0. |Negative root|?

17. x² − 4x − 12 = 0. |Negative root|?

18. x² − 4x − 45 = 0. |Negative root|?

19. x² − 5x − 24 = 0. |Negative root|?

20. x² − 5x − 66 = 0. |Negative root|?

Challenge — 8 Questions

Use the quadratic formula. Give the larger root to 2 d.p.

1. x² − 5x + 3 = 0. Larger root (2 d.p.)?

2. x² − 7x + 3 = 0. Larger root (2 d.p.)?

3. x² − 4x + 1 = 0. Larger root (2 d.p.)?

4. x² − 9x + 5 = 0. Larger root (2 d.p.)?

5. x² − 5x + 2 = 0. Larger root (2 d.p.)?

6. x² − 10x + 8 = 0. Larger root (2 d.p.)?

7. x² − 3x + 1 = 0. Larger root (2 d.p.)?

8. x² − 6x − 2 = 0. Larger root (2 d.p.)?