Quadratic Equations

A quadratic equation looks like ax² + bx + c = 0. The "squared" part is what makes it quadratic. You need to find the values of x that make the equation true — these are called the roots or solutions.

Three Methods to Solve

Factorising — the quickest method when it works. Split into two brackets.
The Quadratic Formula — works for ANY quadratic, even ugly ones.
Completing the Square — useful for finding the turning point of the graph.

Start by trying to factorise. If the numbers are not nice, use the formula.

Key Formulas

Quadratic Formula: x = (−b ± √(b² − 4ac)) / 2a
Discriminant: D = b² − 4ac
D > 0 → two different real roots
D = 0 → one repeated root
D < 0 → no real roots

Worked Examples

Example 1 — Factorising: x² + 5x + 6 = 0

Find two numbers that multiply to give 6 and add to give 5
Those numbers are 2 and 3 (2 × 3 = 6, 2 + 3 = 5)
Write as: (x + 2)(x + 3) = 0
So x + 2 = 0 → x = −2
Or x + 3 = 0 → x = −3

Answer: x = −2 or x = −3

Check: (−2)² + 5(−2) + 6 = 4 − 10 + 6 = 0 ✓

Example 2 — Using the Formula: 2x² − 3x − 5 = 0

Identify: a = 2, b = −3, c = −5
Discriminant: (−3)² − 4(2)(−5) = 9 + 40 = 49
√49 = 7
x = (3 ± 7) / 4
x = (3 + 7)/4 = 10/4 = 2.5
x = (3 − 7)/4 = −4/4 = −1

Answer: x = 2.5 or x = −1

Example 3 — How many roots? x² + 2x + 5 = 0

a = 1, b = 2, c = 5
Discriminant: 2² − 4(1)(5) = 4 − 20 = −16
D < 0, so there are no real roots

Answer: No real solutions

The parabola does not cross the x-axis.

Common Mistakes

Forgetting the equation must equal 0 before factorising
Getting the signs wrong when finding factor pairs
In the formula, forgetting the ± means TWO answers
Not dividing the ENTIRE top by 2a (use brackets!)

Practice Questions

Factorise: x² + 7x + 12 = 0
Factorise: x² − 4x − 21 = 0
Use the formula: x² + 3x − 2 = 0 (give answers to 2 d.p.)
Find the discriminant: 3x² − 2x + 1 = 0. How many roots?
Factorise: 2x² + 5x − 3 = 0

Answers: 1) x = −3 or −4 2) x = 7 or −3 3) x = 0.56 or −3.56 4) D = −8, no real roots 5) x = 0.5 or −3

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