Quadratic equations can be solved by the following methods:

1. factorising
2. graphically
4. discriminant

## Factorising

Look at the National 4 factorising section before continuing.

When a question asks you to 'solve' a quadratic equation, this means that you are to find the roots of the quadratic. In other words, where does the parabola cut the x-axis?

As a graph cuts the axis when the y coordinate is zero, then we substitute $$y = 0$$ into the quadratic equation and use algebra to solve.

### Example

Solve $${x^2} - 9x + 20 = 0$$

We need to factorise the trinomial.

When factorised this is $$(x - 4)(x - 5) = 0$$.

$$(x - 4)$$ and $$(x - 5)$$ are multiplying to give zero, therefore one of these brackets must be equal to zero.

$(x - 4) = 0$

$x = 0 + 4$

$x = 4$

and

$(x - 5) = 0$

$x = 0 + 5$

$x = 5$

Therefore $$x = 4\,and\,x = 5$$ are the roots of quadtratic equations.

Now try the example question below.

Question

Solve $${x^2} + x - 6$$

Factorise the trinomial and then find the two possible $$x$$ values.

${x^2} + x - 6 = 0$

$(x - 2)(x + 3) = 0$

$(x - 2) = 0\,and\,(x + 3) = 0$

$x = 0 + 2 = 2\,and\,x = 0 - 3 = - 3$

Therefore $$x = 2\,and\,x = 3$$