Add the fractions on the left hand side.
When adding/subtracting fractions they must have a common denominator.

Multiply the numerator and denominator of the first fraction by $(x + 1)$ and the numerator and denominator of the second fraction by $x$ .

$\frac{1}{x} \times \frac{(x + 1)}{(x + 1)} - \frac{2}{x + 1} \times \frac{x}{x} = 3$

$\frac{x + 1}{x (x + 1)} - \frac{2 x}{x (x + 1)} = 3$

Common denominator [1]

Subtract the second fraction's numerator from the first and keep over the same denominator.

$\frac{x + 1 - 2 x}{x (x + 1)} = 3$

Simplify the numerator on the left hand side by collecting like terms.

$\frac{1 - x}{x (x + 1)} = 3$

Multiply both sides by $x (x + 1)$ .

$1 - x = 3 x (x + 1)$

[1]

Expand the brackets on the right hand side.

$1 - x = 3 x^{2} + 3 x$

Add $x$ to both sides.

$1 = 3 x^{2} + 4 x$

Subtract 1 from both sides.

$0 = 3 x^{2} + 4 x - 1$

[1]

Use the quadratic formula to solve for $x$ , where $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ given an equation of the form $a x^{2} + b x + c = 0$ .

$x = \frac{- 4 \pm \sqrt{4^{2} - 4 \times 3 \times - 1}}{2 \times 3}$

Correct substitution into the discriminant (the section in the square root) [1]
Correct substitution into the rest of the quadratic formula [1]

$x = \frac{- 4 \pm \sqrt{28}}{6}$

$x = 0.21525 . . .$ or $x = - 1.54858 . . .$

$x = 0.22$ or $x = - 1.55$

0.22 [1]
-1.55 [1]

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Algebraic Fractions (Cambridge O Level Maths)

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