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Maths · Equations and graphs

Quadratic equations

Solve quadratic equations by factorising, by using the quadratic formula and by completing the square, and choose a suitable method.

6 key terms
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Warm-up

Answer each one, then check.

1

Factorise \(x^2 + 5x + 6\).

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\((x + 2)(x + 3)\)
2

Expand \((x + 3)(x - 1)\).

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\(x^2 + 2x - 3\)
3

What is \(\sqrt{49}\)?

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\(7\)
4

If \(ab = 0\), what do you know?

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\(a = 0\) or \(b = 0\)
5

What is the value of \(3^2\)?

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\(9\)

Learning Objectives

1Solve a quadratic by factorising.
2Solve a quadratic with the quadratic formula.
3Complete the square and use it to solve.
4Choose the most efficient method.

QUADRATIC EQUATIONS

A quadratic equation can be written \(ax^2 + bx + c = 0\) and usually has two solutions.

Rearrange so that one side is 0 before you use any method.

Two Solutions

Choosing a Method

Use the quickest one that works.

Factorising
Use it when: The quadratic factorises easily. Example: \(x^2 - 5x + 6 = 0\)
Formula
Use it when: It does not factorise, or the answer is needed as a decimal. Example: \(x^2 + 4x - 7 = 0\)
Completing the square
Use it when: You need exact surds or the turning point. Example: \(x^2 + 6x - 2 = 0\)

Factorising

Solve \(2x^2 + 5x - 3 = 0\).

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1 Factorise \((2x - 1)(x + 3) = 0\)
2 Each bracket is 0 \(2x - 1 = 0\) or \(x + 3 = 0\)
3 Solve \(x = \tfrac{1}{2}\) or \(x = -3\)

Answer\(x = \tfrac{1}{2}\) or \(x = -3\)

The Quadratic Formula

Solve \(x^2 + 4x - 7 = 0\). Give your answers to 2 decimal places.

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1 Formula \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
2 Substitute \(a = 1,\ b = 4,\ c = -7\) \(x = \dfrac{-4 \pm \sqrt{16 + 28}}{2}\)
3 Simplify \(x = \dfrac{-4 \pm \sqrt{44}}{2}\)
4 Evaluate \(x = 1.32\) or \(x = -5.32\)

Answer\(x = 1.32\) or \(x = -5.32\)

Completing the Square

Solve \(x^2 + 6x - 2 = 0\) by completing the square. Give your answers to 2 decimal places.

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1 Complete the square \((x + 3)^2 - 9 - 2 = 0\)
2 Simplify \((x + 3)^2 = 11\)
3 Square root \(x + 3 = \pm\sqrt{11}\)
4 Solve \(x = -3 \pm \sqrt{11} = 0.32\) or \(-6.32\)

Answer\(x = 0.32\) or \(x = -6.32\)

Common Mistakes

Avoid these.

Not rearranging to 0
Move everything to one side first.
Dividing by \(x\)
Solving \(x^2 = 7x\) by dividing loses the solution \(x = 0\); factorise instead.
Signs in the formula
Take care with \(-b\) and \(b^2 - 4ac\) when \(b\) or \(c\) is negative.
Only one solution
Give both solutions.

Choose Your Method

Solve each and say which method you used. (a) \(x^2 - 7x + 10 = 0\) (b) \(x^2 + 2x - 5 = 0\) (2 d.p.) (c) \(x^2 = 9x\).

1. Look for factors first.

2. Otherwise use the formula.

A good answer shows: (a) \(x = 2\) or \(5\) (factorising). (b) \(x = -1 \pm \sqrt{6} = 1.45\) or \(-3.45\) (formula or completing the square). (c) \(x(x - 9) = 0\), so \(x = 0\) or \(9\).

Can I...?

1Rearrange to equal zero.
2Factorise and solve.
3Use the quadratic formula.
4Complete the square.
5Give answers to a given accuracy.
6Give both solutions.
7Avoid dividing by x.
8Choose a method.

Summary & Exam Focus

Factorise if you can.
Formula: \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
Complete the square: \((x + p)^2 + q\).
Every quadratic has up to two solutions.

Exam focus

Solve \(x^2 + 4x - 7 = 0\). Give your solutions correct to 2 decimal places. (3 marks) (3 marks)

Write the formula first, then substitute carefully, using brackets for negatives.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Quadratic: An equation with a highest power of \(x^2\).
Factorise: Write as a product of brackets.
Root: A solution of an equation.
Discriminant: \(b^2 - 4ac\).
Completing the square: Writing \(x^2 + bx + c\) as \((x + p)^2 + q\).
Surd: A root that cannot be simplified to a whole number, e.g. \(\sqrt{11}\).

Practice questions

Have a go at each one before you open its answer.

Question 1 Solve 2 marks

Solve \(x^2 - 5x + 6 = 0\).
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Model answer

\((x - 2)(x - 3) = 0\), so \(x = 2\) or \(x = 3\).

Mark scheme
- \((x - 2)(x - 3)\) — M1
- 2 and 3 — A1
Question 2 Solve 3 marks

Solve \(2x^2 + 5x - 3 = 0\).
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Model answer

\((2x - 1)(x + 3) = 0\), so \(x = \tfrac{1}{2}\) or \(x = -3\).

Mark scheme
- Factorises — M1
- One correct solution — A1
- Both correct — A1
Question 3 Solve 3 marks

Solve \(x^2 + 4x - 7 = 0\). Give your solutions correct to 2 decimal places.
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Model answer

\(x = \dfrac{-4 \pm \sqrt{44}}{2}\); \(x = 1.32\) or \(x = -5.32\).

Mark scheme
- Substitutes into the formula — M1
- \(\dfrac{-4 \pm \sqrt{44}}{2}\) — A1
- 1.32 and \(-5.32\) — A1
Question 4 Solve 4 marks

(a) Write \(x^2 + 6x - 2\) in the form \((x + a)^2 + b\). (b) Hence solve \(x^2 + 6x - 2 = 0\). Give your answers correct to 2 decimal places.
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Model answer

(a) \((x + 3)^2 - 11\). (b) \(x + 3 = \pm\sqrt{11}\), so \(x = 0.32\) or \(x = -6.32\).

Mark scheme
- \(a = 3\) — B1
- \(b = -11\) — B1
- \(x = -3 \pm \sqrt{11}\) — M1
- 0.32 and \(-6.32\) — A1
Question 5 Use the graph 2 marks

The graph of \(y = x^2 - 4x + 1\) is shown. Use the graph to find estimates for the solutions of \(x^2 - 4x + 1 = 0\).
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Model answer

\(x \approx 0.3\) and \(x \approx 3.7\). Accept 0.2 to 0.4 and 3.6 to 3.8.

Mark scheme
- One correct estimate — B1
- Both correct — B1
Question 6 Solve 2 marks

Solve \(x^2 = 7x\).
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Model answer

\(x^2 - 7x = 0\), \(x(x - 7) = 0\), so \(x = 0\) or \(x = 7\).

Mark scheme
- \(x(x - 7) = 0\) — M1
- 0 and 7 — A1

Quick check

How many solutions can a quadratic equation have?
1. AExactly one
2. BExactly three
3. CUp to two
4. DAlways four
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C: Up to two

Up to two.
\((x - 4)(x + 1) = 0\) gives...
1. A\(x = 4\) or \(-1\)
2. B\(x = -4\) or \(1\)
3. C\(x = 4\) only
4. D\(x = -4\) or \(-1\)
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A: \(x = 4\) or \(-1\)

\(x = 4\) or \(x = -1\).
The quadratic formula is...
1. A\(\dfrac{b \pm \sqrt{b^2 + 4ac}}{2a}\)
2. B\(\dfrac{-b \pm \sqrt{b^2 - 4ac}}{a}\)
3. C\(-b \pm \sqrt{b^2 - 4ac}\)
4. D\(\dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
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D: \(\dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

\(\dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
\(x^2 + 6x\) completed the square is...
1. A\((x + 6)^2\)
2. B\((x + 3)^2 - 9\)
3. C\((x + 3)^2 + 9\)
4. D\((x - 3)^2 - 9\)
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B: \((x + 3)^2 - 9\)

\((x + 3)^2 - 9\).
To solve \(x^2 = 9x\) you should...
1. ADivide both sides by \(x\)
2. BSquare root both sides
3. CRearrange to 0 and factorise
4. DGuess
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C: Rearrange to 0 and factorise

Rearrange to \(x^2 - 9x = 0\) and factorise; dividing by \(x\) would lose \(x = 0\).
The solutions of a quadratic are where its graph...
1. ACrosses the x-axis
2. BCrosses the y-axis
3. CHas its turning point
4. DIs steepest
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A: Crosses the x-axis

Crosses the x-axis.

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