Using the Quadratic Formula

# Using the Quadratic Formula

GCSE(H),

The quadratic formula provides the solutions to a quadratic equation. Given ax^2 + bx + c = 0, then the solutions are given by:

x=frac(-b±sqrt(b^2-4ac))(2a)

Note that the squared term is calculated once with a positive value of the squre root; and once with a negative value. Once the solutions have been found, substitute into the original equation to check there has been no arithmetical errors.

Note also that if b^2 < 4ac then there will be an attempt to find the square root of a negative number, and that therefore there are no real roots.

## Examples

1. What, if any, are the solutions for 3x^2-48=0?

Answer: x=+-4

Using the formula, remembering that b=0 as there is no coefficient for x:

x=frac(-(0)+-sqrt((0)^2 - 4(3)(-48)))(2(3)) = +-frac(576)(8) = +-4

Substitute back to check: 3(4)^2 - 48=0 and 3(-4)^2-48=0

2. Determine the solutions for x^2-frac(1)(2)=0. Give your answers in surd form.

Answer: x=+-frac(sqrt(2))(2)

Using the quadratic formula:

x=frac(-(0)+-sqrt((0)^2-4(1)(-frac(1)(2))))(2(1))

x=+-frac(sqrt(2))(2)

Substitute back to check: (frac(sqrt(2))(2))^2 - frac(1)(2)= 0 (true), and (frac(-sqrt(2))(2))^2 - frac(1)(2) = 0 (true).