Solving Quadratic Equations by Factorising

Factorising is the fastest way to solve most quadratic equations. Grade 11 papers use it in almost every algebra and function question, so getting the method automatic saves time on the whole paper.

Standard form

A quadratic equation in standard form is $a x^{2} + b x + c = 0$ , with $a \neq = 0$ .

Always move every term to one side before factorising.

Example. $3 x^{2} + 5 x = 2$ becomes $3 x^{2} + 5 x - 2 = 0$ .

Example. $(x - 3) (x + 1) = 5$ must be expanded first: $x^{2} - 2 x - 3 = 5$ , so $x^{2} - 2 x - 8 = 0$ .

You cannot use the zero product rule when the right-hand side is not 0.

Factorising trinomials

For $x^{2} + b x + c$ , find two numbers that multiply to $c$ and add to $b$ .

$x^{2} - 7 x + 10 = (x - 2) (x - 5)$

For $a x^{2} + b x + c$ with $a \neq = 1$ , try factor pairs of $a$ and $c$ , and check the middle term.

$3 x^{2} + 5 x - 2 = (3 x - 1) (x + 2)$

Check: $3 x \times 2 + (- 1) \times x = 6 x - x = 5 x$ ✓

Take out a common factor first if there is one: $2 x^{2} - 8 x - 10 = 2 (x^{2} - 4 x - 5)$ .

Difference of squares

$a^{2} - b^{2} = (a - b) (a + b)$

Example. $4 x^{2} - 9 = (2 x - 3) (2 x + 3)$ , so $4 x^{2} = 9$ gives $x = \pm \frac{3}{2}$ .

Example. $(2 x - 1)^{2} - (x + 4)^{2} = 0$

$[(2 x - 1) - (x + 4)] [(2 x - 1) + (x + 4)] = 0 \Rightarrow (x - 5) (3 x + 3) = 0$

So $x = 5$ or $x = - 1$ .

The zero product rule

If $A \times B = 0$ , then $A = 0$ or $B = 0$ .

Set each factor equal to 0 and solve.

Never divide by $x$ . In $x^{2} = 5 x$ :

Wrong: divide by $x$ to get $x = 5$ (the solution $x = 0$ is lost)
Right: $x^{2} - 5 x = 0 \Rightarrow x (x - 5) = 0 \Rightarrow x = 0$ or $x = 5$

In context, reject answers that make no sense, such as a negative length.

Key terms

Quadratic equation: An equation where the highest power of the variable is 2.
Trinomial: An expression with three terms.
Difference of squares: An expression of the form $a^{2} - b^{2}$ .
Zero product rule: If a product equals 0, at least one factor is 0.

Common mistakes

Factorising before moving all terms to one side.
Dividing both sides by $x$ and losing $x = 0$ .
Getting the signs wrong in the factors.
Forgetting the negative root of $x^{2} = k$ .

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

3 marks total

knowledge

Solve for $x$ : $x^{2} - 7 x + 10 = 0$

Worked solution

Hidden - try the question first

Solution

$(x - 2) (x - 5) = 0$
$x = 2$
or $x = 5$

Question 2

5 marks total

routine

Solve for $x$ : $3 x^{2} + 5 x = 2$

Worked solution

Hidden - try the question first

Solution

$(x - 2) (x - 5) = 0$
$(3 x - 1) (x + 2) = 0$
$x = \frac{1}{3}$
or $x = - 2$

Question 3

6 marks total

complex

Solve for $x$ :

(a) $(x - 3) (x + 1) = 5$ (3)

(b) $4 x^{2} = 9$ (3)

Worked solution

Hidden - try the question first

Solution

(a) $x^{2} - 2 x - 3 - 5 = 0 \Rightarrow x^{2} - 2 x - 8 = 0$
$(x - 4) (x + 2) = 0$
$x = 4$ or $x = - 2$
(b) $4 x^{2} - 9 = 0$
$(2 x - 3) (2 x + 3) = 0$
$x = \pm \frac{3}{2}$

Question 4

8 marks total

problem-solving

(a) A rectangle is 3 m longer than it is wide. Its area is 40 m². Determine its dimensions. (4)

(b) Thabo solves $x^{2} = 5 x$ by dividing both sides by $x$ and gets $x = 5$ . Explain his mistake and give the full solution. (2)

Worked solution

Hidden - try the question first

Solution

(a) $x^{2} - 2 x - 3 - 5 = 0 \Rightarrow x^{2} - 2 x - 8 = 0$
$x^{2} + 3 x - 40 = 0 \Rightarrow (x + 8) (x - 5) = 0$
$x = 5$ (a width cannot be negative)
5 m by 8 m
(b) Dividing by $x$ loses the solution $x = 0$
$x (x - 5) = 0$ , so $x = 0$ or $x = 5$
(c) $(x - 5) (3 x + 3) = 0$ (difference of squares)
$x = 5$ or $x = - 1$

FAQs

What if a quadratic will not factorise?

Use the quadratic formula or completing the square. If the discriminant is not a perfect square, the trinomial does not factorise over the integers.

Why must the equation equal zero before factorising?

The zero product rule only works when the product equals 0.

Do I need to check answers in context?

Yes. Reject answers that make no sense, such as negative lengths or times.

Standard form

Factorising trinomials

Difference of squares

The zero product rule

Key terms

Common mistakes

Practice questions

FAQs

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