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).

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