Quadratic Equations Step by Step
Learn how to solve quadratic equations by factorising, using the quadratic formula, and completing the square. With clear worked examples at every step.
A quadratic is any equation with an x² in it. The standard form is ax² + bx + c = 0. When you plot it, it makes a U-shape (or an upside-down U if a is negative).
Solving a quadratic means finding where the U crosses the x-axis — the values of x that make the equation equal zero.
Method 1: Factorising
Find two numbers that multiply to give c and add to give b.
Worked Example 1 — Solve x² + 5x + 6 = 0
- Find two numbers that multiply to 6 and add to 5 → 2 and 3
- Write as: (x + 2)(x + 3) = 0
- If the product is 0, one of the brackets must be 0
- x + 2 = 0 → x = −2
- x + 3 = 0 → x = −3
Method 2: The Quadratic Formula
When factorising doesn't work neatly, use the formula:
x = (−b ± √(b² − 4ac)) ÷ 2a
Worked Example 2 — Solve 2x² + 3x − 5 = 0
- a = 2, b = 3, c = −5
- b² − 4ac = 9 − 4(2)(−5) = 9 + 40 = 49
- √49 = 7
- x = (−3 + 7) ÷ 4 = 4 ÷ 4 = 1
- x = (−3 − 7) ÷ 4 = −10 ÷ 4 = −2.5
The Discriminant — How Many Solutions?
The bit under the square root sign, b² − 4ac, is called the discriminant. It tells you how many solutions there are:
| b² − 4ac | What it means |
|---|---|
| > 0 | Two different solutions (curve crosses x-axis twice) |
| = 0 | One repeated solution (curve just touches x-axis) |
| < 0 | No real solutions (curve doesn't reach x-axis) |
Method 3: Completing the Square
Worked Example 3 — Complete the square for x² + 6x + 2 = 0
- Halve the coefficient of x: 6 ÷ 2 = 3
- Write: (x + 3)² − 3² + 2 = 0
- Simplify: (x + 3)² − 9 + 2 = 0
- (x + 3)² = 7
- x + 3 = ±√7
- x = −3 ± √7 ≈ −0.35 or −5.65
Try It Yourself
- Factorise and solve x² − 7x + 12 = 0. (Answer: x = 3 or x = 4)
- Use the quadratic formula on x² + 2x − 8 = 0. (Answer: x = 2 or x = −4)
- What does the discriminant of x² + 4x + 7 = 0 tell you? (Answer: b²−4ac = 16−28 = −12 → no real solutions)
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