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Quadratic Equations

When x² appears, equations curve. Learn to solve quadratics by factorising — and meet the quadratic formula, the tool that never fails.

  • Recognise quadratic equations
  • Solve quadratics by factorising
  • Use the quadratic formula
  • Interpret the two solutions

When x is squared

A quadratic equation contains x² (but no higher powers). Its graph is a curve — a parabola — which is why quadratics can have two solutions: the curve can cross the x-axis twice.

Definition: quadratic equation

A quadratic equation is an equation where the highest power of the unknown is 2, e.g. x² + 5x + 6 = 0 or 2x² − 8 = 0.

Solving by factorising

The key trick: if two things multiply to zero, one of them is zero. So:

  1. Rearrange into the form ... = 0.
  2. Factorise the quadratic.
  3. Set each bracket to zero and solve.
Worked example

Solve x² + 5x + 6 = 0.

Maths
x² + 5x + 6 = 0
(x + 2)(x + 3) = 0      factorise: which pair sums to 5 and multiplies to 6?
x + 2 = 0  or  x + 3 = 0
x = −2  or  x = −3

Check x = −2: 4 − 10 + 6 = 0 ✓

To factorise x² + bx + c, find two numbers that multiply to c and add to b. For x² + 5x + 6: 2 and 3 (2 × 3 = 6, 2 + 3 = 5).

The quadratic formula

Not every quadratic factorises neatly. The quadratic formula solves any quadratic ax² + bx + c = 0:

The quadratic formula

x = (−b ± √(b² − 4ac)) / 2a

Worked example

Solve 2x² + 5x − 3 = 0, where a = 2, b = 5, c = −3.

Maths
x = (−5 ± √(25 − 4(2)(−3))) / 4
x = (−5 ± √(25 + 24)) / 4
x = (−5 ± 7) / 4
x = 0.5  or  x = −3
The discriminant

The part under the square root, b² − 4ac, is called the discriminant. If it's positive there are two solutions; if zero, one repeated solution; if negative, no real solutions.

Practice

Solve: x² − 9 = 0.
Rearrange or factorise as a difference of two squares.

x² = 9, so x = 3 or x = −3. (Or (x − 3)(x + 3) = 0.)

Solve: x² + 7x + 12 = 0 by factorising.
Find two numbers that multiply to 12 and add to 7.

(x + 3)(x + 4) = 0, so x = −3 or x = −4.

Solve x² + 4x + 1 = 0 using the quadratic formula. Give answers to 2 decimal places.
a = 1, b = 4, c = 1.

x = (−4 ± √12)/2 = −2 ± √3, so x ≈ −0.27 or x ≈ −3.73.

Quick check

Quadratic equations — quick check

Which is a quadratic equation?

The highest power of x is 2.

Factorise x² + 8x + 15.

3 × 5 = 15 and 3 + 5 = 8.

Solve (x − 4)(x + 1) = 0.

Each bracket is zero in turn: x = 4 or x = −1.
Key takeaways
  • Quadratics have x² as the highest power and up to two solutions.
  • To factorise: rearrange to = 0, factorise, set each bracket to zero.
  • The quadratic formula x = (−b ± √(b² − 4ac)) / 2a solves any quadratic.
  • The discriminant b² − 4ac tells you how many real solutions exist.
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