GCSE(F), GCSE(H),

There are several ways to work out roots algebraically:

• rearrange as factors; (Factorising)

• the quadratic formula *x* = `frac(-b +- √(b^2 - 4ac))(2a)`; (Quadratic Formula)

• complete the square; (Completing the Square)

• deduce iteratively (for a positive square root).

The roots - and there may be 0, 1 or 2 roots for a quadratic equation - cross the *x*-axis for the solved values of *x*.

Note that the axis of symmetry of a quadratic (and therefore the turning point) will lie halfway between the two roots.

1. Using factors, what are the roots of the function *x*^{2} - 25?

Answer: (5, 0) and (-5, 0)

Factorising the equation gives (*x* - 5)(*x* + 5); therefore *x* = -5 and *x* = +5 are the roots (make each of the brackets equal to zero, in turn).

2. By completing the square, find the roots of *x*^{2} - 8*x* + 12 = 0.

Answer: (6, 0) and (2, 0)

The integer part of the squared term is given by `frac(b)(2)`;

The squared term is therefore (*x* + `frac(-8)(2)`)

(*x* - 4)^{2} = *x*^{2} - 8*x* + 16

(*x* - 4)^{2} - 4 = *x*^{2} - 8*x* + 12

Roots are given by (*x* - 4)^{2} - 4 = 0

(*x* - 4)^{2} = 4

*x* - 4 = +/- 2, so *x* = 6 or 2.

Our iOS app has over 1,000 questions to help you practice this and many other topics.

Available to download free on the App Store.

Available to download free on the App Store.