To solve a quadratic equation, rearrange it to ax² + bx + c = 0. Then factorise and set each bracket equal to zero, or use the quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a. A quadratic can have two, one or no real solutions.
The key idea
Always get zero on one side first. Factorising works because if two numbers multiply to make zero, one of them must be zero.
The discriminant b² − 4ac tells you how many real solutions there are: positive means two, zero means one, negative means none.
Solving means finding where the graph crosses the x-axis. The graph of y = x² + 5x + 6 is a U-shaped curve. It crosses the x-axis at x = −3 and x = −2, so those are the solutions of x² + 5x + 6 = 0.
A quadratic can cross twice (two solutions), touch once (one solution) or never reach the axis (no real solutions).
Worked examples
Example 1Solve x² + 5x + 6 = 0.
Factorise: two numbers that multiply to 6 and add to 5
- x² + 5x + 6 = 0× to 6, + to 5
- 2 × 3 = 6, 2 + 3 = 5The numbers are 2 and 3
- (x + 2)(x + 3) = 0Write as two brackets
- x + 2 = 0 or x + 3 = 0One bracket must equal zero
- x = −2 or x = −3Check: 4 − 10 + 6 = 0 ✓
Answer: x = −2 or x = −3
Example 2Solve 2x² − 3x − 4 = 0.
The quadratic formula, with a = 2, b = −3, c = −4
- x = (−b ± √(b² − 4ac)) ÷ 2aWrite the formula
- x = (3 ± √(9 + 32)) ÷ 4−(−3) = 3; −4 × 2 × (−4) = +32
- x = (3 ± √41) ÷ 4√41 = 6.403…
- x = 9.403… ÷ 4 or x = −3.403… ÷ 4
- x ≈ 2.35 or x ≈ −0.8513 significant figures
Answer: x ≈ 2.35 or x ≈ −0.851
Example 3A ball’s height in metres after t seconds is h = 20t − 5t². When does it land?
Set the height to zero and factorise
- 20t − 5t² = 0It lands when h = 0
- 5t(4 − t) = 0Common factor 5t
- 5t = 0 or 4 − t = 0One factor must be zero
- t = 0 or t = 4t = 0 is the throw; it lands after 4 s
Answer: After 4 seconds
Common mistakes
- Trying to factorise before rearranging to “= 0”.
- Dividing both sides by x and losing a solution: x² = 3x has two solutions, x = 0 and x = 3.
- Sign errors with −b in the formula when b is negative.
Practice questions
- x² − 7x + 12 = 0
Show answer
x = 3 or x = 4
- x² + 2x − 15 = 0
Show answer
x = −5 or x = 3
- x² = 5x
Show answer
x = 0 or x = 5
- x² + 4x + 1 = 0 (to 3 significant figures)
Show answer
x = −2 ± √3, so x ≈ −0.268 or x ≈ −3.73
More topic explainers
- Solving linear equations (MYP 2–3)
- Pythagoras’ theorem (MYP 3)
- Right-angled trigonometry (SOH CAH TOA) (MYP 4)
Find more free resources, or look up command terms in the glossary.