Year 10: Solve the quadratic equation by factorisation

What is factorisation?

Factorisation involves rewriting the quadratic expression into the form $(x+p)(x+q)=0$, where p and q are constants.

Steps to solve by factorisation

1. Rearrange the equation: Make sure the equation is in the form $ax\mathrm{²}+bx+c=0$.
2. Factorise the quadratic expression: Find two numbers that multiply to 'ac' (a multiplied by c) and add up to 'b'.
3. Rewrite the equation: Replace the 'bx' term with the sum of those two numbers.
4. Factorise the new expression: You should now be able to factorise the entire expression into two binomial factors.
5. Set each factor equal to zero:$(x+p)(x+q)=0$ implies $x+p=0$ and $x+q=0$.
6. Solve for x: Solve each of the two linear equations to find the values of x.

Example

Solve: $x\mathrm{²}+5x+6=0$

1. Find two numbers that multiply to 6 and add to 5. These are 2 and 3.
2. Rewrite the equation: $x\mathrm{²}+2x+3x+6=0$
3. Factorise: $x(x+2)+3(x+2)=0$
4. Rewrite: $(x+2)(x+3)=0$
5. Solve: $x+2=0\Rightarrow x=-2$ and $x+3=0\Rightarrow x=-3$

Your solutions are $x=-2$ and $x=-3$.