The Quadratic Formula

Q: How do I know if a quadratic will factorise?

If $b^2 - 4ac$ is a perfect square, it factorises over the rational numbers.

The quadratic formula solves every quadratic equation, whether it factorises or not. Grade 11 questions test accurate substitution, answers in the right form, and judgement about when factorising is quicker.

Using the quadratic formula

For $a x^{2} + b x + c = 0$ :

$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$

Write the equation in standard form.
Identify $a$ , $b$ and $c$ , with their signs.
Substitute, using brackets for negative values.

Example. $x^{2} - 4 x - 1 = 0$ : $a = 1$ , $b = - 4$ , $c = - 1$ .

$x = \frac{4 \pm 16 + 4}{2} = \frac{4 \pm 2 5}{2} = 2 \pm 5$

Answers in simplest surd form or rounded

Read the question for the form of the answer.

Simplest surd form: simplify the root and cancel common factors.

$x = \frac{- 2 \pm 52}{6} = \frac{- 2 \pm 2 13}{6} = \frac{- 1 \pm 13}{3}$

Rounded: use your calculator at the end only. $x \approx 0, 87$ or $x \approx - 1, 54$ .

Cancel only when every term in the numerator shares the factor.

Choosing the best method

Factorising is quickest when the trinomial factorises easily, especially for perfect squares like $4 x^{2} - 12 x + 9 = (2 x - 3)^{2}$ .
The formula always works, and is best when the answers are irrational or the question asks for decimals.
Completing the square is best for finding turning points.

Check $b^{2} - 4 a c$ first: if it is a perfect square, the trinomial factorises.

Example in context. A ball's height is $h = - 5 t^{2} + 20 t + 2$ metres. It is 12 m high when $- 5 t^{2} + 20 t - 10 = 0$ , so $t^{2} - 4 t + 2 = 0$ and $t = 2 \pm 2$ : at about 0,59 s and 3,41 s.

Key terms

Quadratic formula: $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ .
Coefficient: The number multiplying a variable term.
Simplest surd form: An answer with the surd simplified and common factors cancelled.

Common mistakes

Writing $- b$ as $- 4$ when $b = - 4$ ; it should be $+ 4$ .
Dividing only part of the numerator by $2 a$ .
Cancelling a factor from one term of the numerator but not the other.
Forgetting to rewrite the equation in standard form first.

Practice questions

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

Question 1

3 marks total

knowledge

Solve $x^{2} - 4 x - 1 = 0$ . Leave your answer in simplest surd form.

Worked solution

Hidden - try the question first

Solution

$x = \frac{4 \pm 16 + 4}{2}$
$= \frac{4 \pm 2 5}{2}$
$= 2 \pm 5$

Question 2

5 marks total

routine

Solve $3 x^{2} + 2 x - 4 = 0$ . Give your answers in simplest surd form and to two decimal places.

Worked solution

Hidden - try the question first

Solution

$x = \frac{4 \pm 16 + 4}{2}$
$= \frac{- 2 \pm 2 13}{6}$
$= \frac{- 1 \pm 13}{3}$
$x \approx 0, 87$
or $x \approx - 1, 54$

Question 3

6 marks total

complex

(a) Solve $x (x - 3) = 2 (x + 1)$ , correct to two decimal places. (4)

(b) Which method is best for $4 x^{2} - 12 x + 9 = 0$ ? Solve it. (2)

Worked solution

Hidden - try the question first

Solution

(a) $x^{2} - 3 x = 2 x + 2 \Rightarrow x^{2} - 5 x - 2 = 0$
$x = \frac{5 \pm 25 + 8}{2}$
$x \approx 5, 37$
or $x \approx - 0, 37$
(b) Factorising: $(2 x - 3)^{2} = 0$
$x = \frac{3}{2}$

Question 4

8 marks total

problem-solving

A ball is kicked upwards. Its height after $t$ seconds is $h = - 5 t^{2} + 20 t + 2$ metres.

(a) After how many seconds does the ball hit the ground? Give your answer to two decimal places. (4)

(b) At what times is the ball 12 m above the ground? Give exact answers and decimals. (4)

Worked solution

Hidden - try the question first

Solution

(a) $x^{2} - 3 x = 2 x + 2 \Rightarrow x^{2} - 5 x - 2 = 0$
$t = \frac{20 \pm 400 + 40}{10}$
$t \approx 4, 10$ s or $t \approx - 0, 10$ s
Reject the negative time: $t \approx 4, 10$ s
(b) $- 5 t^{2} + 20 t + 2 = 12 \Rightarrow t^{2} - 4 t + 2 = 0$
$t = \frac{4 \pm 8}{2}$
$t = 2 \pm 2$
$t \approx 0, 59$ s and $t \approx 3, 41$ s

FAQs

Is the quadratic formula given in the exam?

Yes, it is on the formula sheet, but you need to use it quickly and accurately.

When should I give answers in surd form?

When the question asks for exact answers or simplest surd form. Otherwise round as instructed.

How do I know if a quadratic will factorise?

If $b^{2} - 4 a c$ is a perfect square, it factorises over the rational numbers.

Using the quadratic formula

Answers in simplest surd form or rounded

Choosing the best method

Key terms

Common mistakes

Practice questions

FAQs

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