Quadratic equations have one variable and the highest exponent is 2.

Quadratic equations can have zero, one or two solutions. At this level, there are three methods for solving quadratic equations.

### By factorising.

Rearrange the equation so that all of the terms are on one side of the equation. Then factorise.

The method depends on the fact that if two factors multiplied together are equal to zero, then either one or both of them must be equal to zero.

 Examples Answers Solve: (a) x2 − 4x − 5 = 0 Factorise to (x − 5)(x + 1) = 0 Either: x − 5 = 0 or x + 1 = 0 x = 5 or x = -1 (b) (2x − 3)(x + 4) = 0 Already factorised Either: 2x − 3 = 0 or x + 4 = 0 2x = 3 or x = − 4 x = 1.5 (c) 3x2 − 6x = 0 Factorise to 3x(x − 2) = 0 Either: 3x = 0 or x − 2 = 0 x = 0 or x = 2 (d) (x − 4)2 = 0 Already factorised (x − 4)(x − 4) = 0 Either: x − 4 = 0 or x − 4 = 0 x = 4

### By taking the square root.

Take the square root of both sides of the equation.

Remember, if x2 = a then Examples Answers (a) Solve x2 = 36 Take the square root of both sides. x = ± 6 (b) Solve (x − 4)2 = 9 Take the square root of both sides. x − 4 = ± 3 x = 7 or 1

### By graphing.

The solutions for the equation are the x-intercepts of the parabola. See Topic 38