The two fractions are being multiplied and involve three quadratic expressions.
Factorise these expressions as this will make them easier to multiply.

$\begin{array}{rcl} 25 x^{2} - 64 & = & (5 x - 8) (5 x + 8) \\ 5 x^{2} - 13 x - 6 & = & (5 x + 2) (x - 3) \\ x^{2} - 8 x + 15 & = & (x - 5) (x - 3) \end{array}$

One quadratic correctly factorised [1]
Two quadratics correctly factorised [1]

Rewrite the product of the fractions.

$\frac{25 x^{2} - 64}{5 x^{2} - 13 x - 6} \times \frac{x^{2} - 8 x + 15}{5 x + 8} = \frac{(5 x - 8) (5 x + 8)}{(5 x + 2) (x - 3)} \times \frac{(x - 5) (x - 3)}{(5 x + 8)}$

Look to cancel any factors before multiplying.

$\frac{(5 x - 8) (5 x + 8)}{(5 x + 2) (x - 3)} \times \frac{(x - 5) (x - 3)}{(5 x + 8)} = \frac{(5 x - 8) (x - 5)}{(5 x + 2)}$

Now we need to include the third term, $- (x - 7)$ , and make a common denominator.

$\frac{(5 x - 8) (x - 5)}{(5 x + 2)} - (x - 7) = \frac{(5 x - 8) (x - 5) - (x - 7) (5 x + 2)}{(5 x + 2)}$

[1]

Expand and simplify the numerator.

$\begin{array}{rcl} \frac{(5 x - 8) (x - 5) - (x - 7) (5 x + 2)}{(5 x + 2)} & = & \frac{5 x^{2} - 33 x + 40 - (5 x^{2} - 33 x - 14)}{(5 x + 2)} \\ = & \frac{5 x^{2} - 33 x + 40 - 5 x^{2} + 33 x + 14}{(5 x + 2)} \\ = & \frac{54}{(5 x + 2)} \end{array}$

Nothing else can be simplified or cancelled.

$\frac{25 x^{2} - 64}{5 x^{2} - 13 x - 6} \times \frac{x^{2} - 8 x + 15}{5 x + 8} = \frac{54}{5 x + 2}$ [1]

	Distance (miles)	Speed (mph)	Time (hours)
1st stage	731	$x$	$\frac{731}{x}$
2nd stage	287	$x - 24$	$\frac{287}{x - 24}$

Algebraic Fractions (AQA GCSE Maths)

Exam Questions

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

2. Algebra

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Algebraic Roots & Indices

Expanding Brackets

Factorising

Completing the Square

Algebraic Fractions

Rearranging Formulae

Algebraic Proof

Linear Equations

Solving Quadratic Equations

Simultaneous Equations

Iteration

Forming & Solving Equations

Functions

Coordinate Geometry

Linear Graphs y = mx + c

Graphs of Functions

Equation of a Circle

Estimating Gradients & Areas under Graphs

Real-Life Graphs

Solving Inequalities

Graphing Inequalities

Transformations of Graphs

Sequences

3. Ratio, Proportion & Rates of Change

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Ratio Problem Solving

Direct & Inverse Proportion

Standard & Compound Units

Exchange Rates & Best Buys

4. Geometry & Measures

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Angles in Polygons & Parallel Lines

Bearings, Scale Drawing, Constructions & Loci

Circle Theorems

Area & Perimeter

Circles, Arcs & Sectors

Volume & Surface Area

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Area & Volume of Similar Shapes

Right-Angled Triangles - Pythagoras & Trigonometry

Sine, Cosine Rule & Area of Triangles

3D Pythagoras & Trigonometry

Vectors

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5. Probability

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6. Statistics

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