12.1 Expanding and factorising quadratics — Learn

Prerequisites: collecting like terms, multiplying terms, and taking out a common factor.

Objectives: expand two brackets and reverse the process to factorise a monic quadratic. These methods cover A4 and the expression work in A18.

Expand two brackets

Multiply each term in the first bracket by each term in the second. Then collect like terms.

Example 1. Expand $(x+3)(x+5)$.

The middle terms combine because both contain $x$.

Factorise a quadratic

A quadratic expression has a highest power of $x$ equal to two. To factorise $x^2+bx+c$, find two numbers whose sum is $b$ and product is $c$.

Example 2. Factorise $x^2+7x+12$.

The numbers $3$ and $4$ add to $7$ and multiply to $12$.

Expanding checks the answer: $x^2+4x+3x+12=x^2+7x+12$.

Solve by factorising

To solve a quadratic equation, first make one side zero. Then factorise and set each factor equal to zero.

Example 3. Solve $x^2+5x+6=0$.

So $x+2=0$ or $x+3=0$. Therefore $x=-2$ or $x=-3$.

Watch out. The factors must have both the correct sum and product. For $x^2+7x+12$, $(x+2)(x+6)$ has the right product, but its middle coefficient is $8$. Expand to check.

Progress check

The height model for the chapter problem is $h=-t^2+4t+5$. Notice its quadratic term and two other terms. Later, factorising will help identify when the height is zero.

You should now be able to

• multiply every term in one bracket by every term in another;
• collect like terms after expansion;
• factorise a monic quadratic using the sum and product of two numbers;
• solve a factorisable quadratic equation by setting each factor to zero;
• check a factorisation by expanding it.