Working with Roots in Algebra

# Working with Roots in Algebra

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.

## Examples

1. Using factors, what are the roots of the function x2 - 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 x2 - 8x + 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 = x2 - 8x + 16

(x - 4)2 - 4 = x2 - 8x + 12

Roots are given by (x - 4)2 - 4 = 0

(x - 4)2 = 4

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