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# Solving Equations by Factorising

### Solving Equations by Factorising

Factorising is the first step in solving an equation, which we can use to find the value of an unknown quantity.

Factorising can be the first step to solving a quadratic equation to find the value of an unknown.

When solving equations by factorising, we often end up with two sets of brackets equal to zero. This means that we find **two solutions for** $x$.

Let's start with a quadratic: $x^2-7x+12=0$

First of all, we need to factorise the equation

We can factorise by finding a pair of factors which multiply to make $12$, and add together to give $-7$.

Factorise $x^2-7x+12=0$

We can factorise to form $(x - 3)(x - 4) = 0$

This means that we have **two** solutions for $x$, one in each bracket. So, we effectively have $x-3=0$ and $x-4=0$

For $x-3=0$, subtract 3 from both sides

$x\space\cancel{-3}=0+3$ so $x=3$

If $x-4=0$, what is $x$?

$x=4$ when $x-4=0$

Therefore, there are two solutions to the equation. $x=3$ and $x=4$.

What are the solutions to the equation: $(x - 5)(x - 6) = 0$?

Find the two solutions to the equation: $(x - 4)(x + 7) = 0$

You may have to do some more work to find the values of $x$ in some cases.

Let's try solving $(2x + 3)(3x - 1) = 0$

We can treat this as two equations

We need to solve $2x+3=0$ and $3x-1=0$. This means that we will have two different solutions for $x$.

Solve the left bracket $2x+3=0$

Solve the right bracket $3x-1=0$

There we go! You've solved it

$x=-\dfrac{3}{2}$ and $x=\dfrac{1}{3}$

Always give these answers as exact fractions where possible

Don't give mixed numbers!

Solve $(2x + 4)(3x - 12) = 0$

What are the solutions to the equation: $(5x + 3)(7x - 24) = 0$?