The ratios are equivalent to each other, this can be easiest to look at by writing one above the other.

$\begin{array}{rcl} 2 x - 1 : x - 4 \\ 16 x + 1 : 2 x - 1 \end{array}$

As the ratios are equal, their scale factors must also be equal.

Find the scale factors by looking at the expressions on each side of the ratios separately.
Considering the left-hand side and letting the scale factor be a.

$\begin{array}{rcl} (2 x - 1) a & = & 16 x + 1 \\ a & = & \frac{16 x + 1}{2 x - 1} \end{array}$

Considering the right-hand side and letting the scale factor be a.

$\begin{array}{rcl} (x - 4) a & = & 2 x - 1 \\ a & = & \frac{2 x - 1}{x - 4} \end{array}$

Setting the expressions found for a equal to each other gives:

$\begin{array}{rcl} \frac{16 x + 1}{2 x - 1} & = & \frac{2 x - 1}{x - 4} \end{array}$

[1]

Remove the fractions by multiplying each side by each expression on the denominator.
Remember to use brackets when multiplying by an expression.

$\begin{array}{rcl} (x - 4) (2 x - 1) \frac{16 x + 1}{2 x - 1} & = & \frac{2 x - 1}{x - 4} (x - 4) (2 x - 1) \\ (x - 4) (16 x + 1) & = & (2 x - 1) (2 x - 1) \end{array}$

[1]

Expand the brackets on both sides.

$\begin{array}{rcl} 16 x^{2} + x - 64 x - 4 & = & 4 x^{2} - 2 x - 2 x + 1 \end{array}$

Simplify.

$\begin{array}{rcl} 16 x^{2} - 63 x - 4 & = & 4 x^{2} - 4 x + 1 \end{array}$

Rearrange to make a quadratic equation equal to zero.

$\begin{array}{rcl} 12 x^{2} - 59 x - 5 & = & 0 \end{array}$

[1]

Solve by substituting the values into the quadratic formula.
The quadratic formula is $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ where $a x^{2} + b x + c = 0$ .

$a = 12, b = - 59, c = - 5$

$x = \frac{- (- 59) \pm \sqrt{{(- 59)}^{2} - 4 \times 12 \times (- 5)}}{2 \times 12}$

[1]

$x = \frac{59 \pm \sqrt{3721}}{24}$

$x = 5 or x = - \frac{1}{12}$ [1]

Solving Quadratic Equations (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