Quadratic Equation Methods

If you have to solve a quadratic equation but are not told which method to use, here is a guide as to what to do

When should I solve by factorisation?

When the question asks to solve by factorisation
- For example, part (a) Factorise 6x² + 7x – 3, part (b) Solve 6x² + 7x – 3 = 0
When solving two-term quadratic equations
- For example, solve x² – 4x = 0
  - …by taking out a common factor of x to get x(x – 4) = 0
  - ...giving x = 0 and x = 4
- For example, solve x² – 9 = 0
  - …using the difference of two squares to factorise it as (x + 3)(x – 3) = 0
  - ...giving x = -3 and x = 3
  - (Or by rearranging to x² = 9 and using ±√ to get x = = ±3)

When should I use the quadratic formula?

When the question says to leave solutions correct to a given accuracy (2 decimal places, 3 significant figures etc)
When the quadratic formula may be faster than factorising
- It's quicker to solve 36x² + 33x – 20 = 0 using the quadratic formula then by factorisation
If in doubt, use the quadratic formula - it always works

When should I solve by completing the square?

When part (a) of a question says to complete the square and part (b) says to use part (a) to solve the equation
When making x the subject of harder formulae containing x² and x terms
- For example, make x the subject of the formula x² + 6x = y
  - Complete the square: (x + 3)² – 9 = y
  - Add 9 to both sides: (x + 3)² = y + 9
  - Take square roots and use ±: $x + 3 = \pm \sqrt{y + 9}$
  - Subtract 3: $x = - 3 \pm \sqrt{y + 9}$

Exam Tip

Calculators can solve quadratic equations so use them to check your solutions
If the solutions on your calculator are whole numbers or fractions (with no square roots), this means the quadratic equation does factorise

Worked example

(a)

Solve

x^{2} - 7 x + 2 = 0

, giving your answers correct to 2 decimal places

“Correct to 2 decimal places” suggests using the quadratic formula
Substitute a = 1, b = -7 and c = 2 into the formula, putting brackets around any negative numbers

$x = \frac{- (- 7) \pm \sqrt{{(- 7)}^{2} - 4 \times 1 \times 2}}{2 \times 1}$

Use a calculator to find each solution

x = 6.70156… or 0.2984...

Round your final answers to 2 decimal places

x = 6.70 or x = 0.30

(b)

Solve

16 x^{2} - 82 x + 45 = 0

Method 1
If you cannot spot the factorisation, use the quadratic formula
Substitute a = 16, b = -82 and c = 45 into the formula, putting brackets around any negative numbers

$x = \frac{- (- 82) \pm \sqrt{{(- 82)}^{2} - 4 \times 16 \times 45}}{2 \times 16}$

Use a calculator to find each solution

x = $\frac{9}{2}$ or x = $\frac{5}{8}$

Method 2
If you do spot the factorisation, (2x – 9)(8x – 5), then use that method instead

$(2 x - 9) (8 x - 5) = 0$

Set the first bracket equal to zero

$2 x - 9 = 0$

Add 9 to both sides then divide by 2

$\begin{array}{rcl} 2 x & = & 9 \\ x & = & \frac{9}{2} \end{array}$

Set the second bracket equal to zero

$8 x - 5 = 0$

Add 5 to both sides then divide by 8

$\begin{array}{rcl} 8 x & = & 5 \\ x & = & \frac{5}{8} \end{array}$

x = $\frac{9}{2}$ or x = $\frac{5}{8}$

(c)

By writing

x^{2} + 6 x + 5

in the form

{(x + p)}^{2} + q

, solve

x^{2} + 6 x + 5 = 0

This question wants you to complete the square first
Find p (by halving the middle number)

$p = \frac{6}{2} = 3$

Write x² + 6x as (x + p)² - p²

$\begin{array}{rcl} x^{2} + 6 x & = & {(x + 3)}^{2} - 3^{2} \\ = & {(x + 3)}^{2} - 9 \end{array}$

Replace x² + 6x with (x + 3)² – 9 in the equation

$\begin{array}{rcl} {(x + 3)}^{2} - 9 + 5 & = & 0 \\ {(x + 3)}^{2} - 4 & = & 0 \end{array}$

Make x the subject of the equation (start by adding 4 to both sides)

${(x + 3)}^{2} = 4$

Take square roots of both sides (include a ± sign to get both solutions)

$x + 3 = \pm \sqrt{4} = \pm 2$

Subtract 3 from both sides

$x = \pm 2 - 3$

Find each solution separately using + first, then - second

x = - 5, x = - 1

Even though the quadratic factorises to (x + 5)(x + 1), this is not the method asked for in the question

1. Number

Number Toolkit

Prime Factors, HCF & LCM

Powers, Roots & Standard Form

Fractions

Percentages

Simple & Compound Interest, Growth & Decay

Fractions, Decimals & Percentages

Rounding, Estimation & Bounds

Surds

Using a Calculator

Using a Calculator

2. Algebra

Algebra Toolkit

Algebraic Roots & Indices

Algebraic Roots & Indices

Expanding Brackets

Factorising

Completing the Square

Completing the Square

Rearranging Formulae

Rearranging Formulae

Algebraic Proof

Algebraic Proof

Linear Equations

Solving Linear Equations

Solving Quadratic Equations

Simultaneous Equationsr

Simultaneous Equations

Iteration

Iteration

Forming & Solving Equations

Functions

Coordinate Geometrys

Linear Graphs y = mx + c

Graphs of Functions

Equation of a Circle

Estimating Gradients & Areas under Graphs

Real-Life Graphs

Solving Inequalities

Graphing Inequalities

Graphing Inequalities

Transformations of Graphs

Reflections of Graphs

Sequences

3. Ratio, Proportion & Rates of Change

Ratio Toolkit

Ratio Problem Solving

Multiple Ratios

Direct & Inverse Proportions

Direct & Inverse Proportion

Standard & Compound Units

Exchange Rates & Best Buys

Exchange Rates & Best Buys

4. Geometry & Measures

Geometry Toolkit

Angles in Polygons & Parallel Lines

Bearings, Scale Drawing, Constructions & Loci

Circle Theorems

Area & Perimeter

Circles, Arcs & Sectors

Volume & Surface Area

Congruence, Similarity & Geometrical Proof

Area & Volume of Similar Shapes

Similar Area & Volumes

Right-Angled Triangles – Pythagoras & Trigonometry

Sine, Cosine Rule & Area of Triangles

3D Pythagoras & Trigonometry

3D Pythagoras & Trigonometry

Vectors

Transformations

5. Probability

Probability Toolkit

Simple Probability Diagrams

Tree Diagrams

Tree Diagrams

Combined & Conditional Probability

6. Statistics

Statistics Toolkit

Statistical Diagrams

Histograms

Histograms

Cumulative Frequency & Box Plots

Scatter Graphs & Correlation

Scatter Graphs

Top Rated by Parents/Students Nationwide