12.4 Solving quadratic equations by factorising — Learn

Prerequisites: factorising monic quadratics and solving linear equations.

Objectives: solve quadratic equations by factorising and using the zero-product rule. This covers A18.

Use the zero-product rule

If two factors multiply to zero, at least one factor must be zero. Factorise the equation, set each factor to zero, and solve.

Example 1. Solve $x^2+5x+6=0$.

$$ (x+2)(x+3)=0 $$ So $x+2=0$ or $x+3=0$.

Example 2. Solve $x^2=5x+14$.

$$ (x-7)(x+2)=0 $$ Therefore $x=7$ or $x=-2$.

Watch out. From $ab=0$, do not divide by $a$ or $b$. Either factor could be zero, so set each factor equal to zero.

You should now be able to

• rearrange a quadratic equation so one side is zero;
• factorise and apply the zero-product rule;
• give both solutions and check them in the original equation.