Method 1 - Decimal search

The equation already has 0 on one side; $x^{4} - x^{2} - 9 = 0$ .
Substitute 1.0, 1.1, 1.2, 1.3 ... into the equation until a change of sign happens.

$x$	$x^{4} - x^{2} - 9$
1.0	-9
1.1	-8.7459
1.2	-8.3664
1.3	-7.8339
1.4	-7.1184
1.5	-6.1875
1.6	-5.0064
1.7	-3.5379
1.8	-1.7424
1.9	0.4221

One correct evaluation between 1 and 2 [1]
Two correct evaluations between 1 and 2 [1]

There is a sign change between 1.8 and 1.9. So we now test the midpoint, 1.85

when $x = 1.85, x^{4} - x^{2} - 9 = - 0.708 993 . . .$

[1]

Therefore the sign change is between 1.85 and 1.9 so the solution will round to 1.9 to one decimal place.

$x = 1.9 (1 d . p .)$ [1]

Method 2 - Interval bisection

The equation already has 0 on one side; $x^{4} - x^{2} - 9 = 0$ .
1.5 bisects the interval 1 to 2.
Substitute 1, 1.5 and 2 into the (left-hand side of the) equation.

$x$	$x^{4} - x^{2} - 9$
1	-9
1.5	-6.1875
2	3

First correct evaluation between 1 and 2 [1]

There is a change of sign between 1.5 and 2. Bisect this interval (1.75) and test this value in the equation.

when $x = 1.75, x^{4} - x^{2} - 9 = - 2.683 593 . . .$

Second correct evaluation between 1 and 2 [1]

There is a change of sign between 1.75 and 2. Bisect this interval (1.875) and test this value in the equation.

when $x = 1.875, x^{4} - x^{2} - 9 = - 0.156 005 . . .$

There is a change of sign between 1.875 and 2. Bisect this interval ( (1.875 + 2) ÷ 2 = 1.9375) and test this value in the equation.

when $x = 1.9375, x^{4} - x^{2} - 9 = 1.337 905 . . .$

[1]

There is a change of sign between 1.875 and 1.9375 so the solution will round to 1.9 to one decimal place.

$x = 1.9 (1 d . p .)$ [1]

Iteration (OCR GCSE Maths)

Exam Questions

1. Number Operations & Integers

Number Toolkit

Prime Factors, HCF & LCM

2. Fractions, Decimals & Percentages

Fractions

Percentages

Working with FDP

3. Indices & Surds

Powers, Roots & Standard Form

Surds

4. Approximation & Estimation

Rounding, Estimation & Bounds

Using a Calculator

5. Ratio, Proportion & Rates of Change

Ratio Toolkit

Ratio Problem Solving

Direct & Inverse Proportion

Exchange Rates & Best Buys

Simple & Compound Interest, Growth & Decay

6. Algebra

Algebra Toolkit

Algebraic Roots & Indices

Expanding Brackets

Factorising

Completing the Square

Algebraic Fractions

Rearranging Formulae

Algebraic Proof

Linear Equations

Solving Quadratic Equations

Solving Inequalities

Simultaneous Equations

Iteration

Forming & Solving Equations

Functions

Sequences

7. Graphs of Equations & Functions

Coordinate Geometry

Linear Graphs y = mx + c

Graphing Inequalities

Graphs of Functions

Equation of a Circle

Estimating Gradients & Areas under Graphs

Real-Life Graphs

Transformations of Graphs

8. Basic Geometry

Geometry Toolkit

Angles in Polygons & Parallel Lines

Circle Theorems

9. Congruence & Similarity

Congruence, Similarity & Geometrical Proof

Area & Volume of Similar Shapes

Vectors

Transformations

10. Mensuration

Standard & Compound Units

Bearings, Scale Drawing, Constructions & Loci

Area & Perimeter

Circles, Arcs & Sectors

Volume & Surface Area

Right-Angled Triangles - Pythagoras & Trigonometry

Sine, Cosine Rule & Area of Triangles

3D Pythagoras & Trigonometry

11. Probability

Probability Toolkit

Simple Probability Diagrams

Tree Diagrams

Combined & Conditional Probability

12. Statistics

Statistics Toolkit

Statistical Diagrams

Histograms

Cumulative Frequency & Box Plots

Scatter Graphs & Correlation