The Quadratic Formula

The quadratic formula gives the solutions to any quadratic equation. It can be used to solve quadratic equations which cannot be solved by factorisation using rational numbers.

Before we can use the quadratic formula, we have to rearrange the quadratic equation into the form $ax^2+bx+c=0$, where $a$, $b$, and$c$ are any number and $a\ne0$. Once the equation is in this form, the solutions are given by the quadratic formula:

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$a$ is the coefficient of $x^2$

$b$ is the coefficient of $x$

$c$ is the constant

The Discriminant

A critical component is called the discriminant: ∆ = $b^2-4ac$

It discriminates whether factorisation is possible, and determines the number and type of real solutions a quadratic function will have.

If $b^2-4ac\ge0$ then we can factorise a quadratic.

If b^2-4ac<0 then we cannot factorise a quadratic, as the value under the square root is negative.

For quadratics equal to 0…

If ∆ >0 then there are 2 real solutions.

If ∆ $=0$ then there is 1 real solution.

If ∆ <0 then there are no real solutions