Teach Me Maths

Algebraic Fractions

This section is in development. More questions and features coming shortly.

AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen and paper handy, to work out your answers.

Question 13 marks

Simplify fully

\dfrac{3x^2-12}{x^2+5x+6}

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Factorise the top and the bottom completely before you cancel anything: the top has a common factor and then a difference of two squares.

Worked solution
  1. Top: take out the common factor 3: 3x^2-12=3(x^2-4)
  2. x^2-4 is a difference of two squares: 3(x^2-4)=3(x-2)(x+2)
  3. Bottom: x^2+5x+6=(x+2)(x+3)
  4. \dfrac{3(x-2)(x+2)}{(x+2)(x+3)}
  5. Cancel the common factor (x+2)
  6. Answer: \dfrac{3(x-2)}{x+3}

Question 21 mark

Which of these is equal to \dfrac{2}{x+1}-\dfrac{1}{x} ?

Select the correct answer.

Hint

Use the common denominator x(x+1) and multiply each numerator by whatever its denominator was multiplied by.

Worked solution
  1. Common denominator: x(x+1)
  2. \dfrac{2}{x+1}=\dfrac{2x}{x(x+1)} and \dfrac{1}{x}=\dfrac{x+1}{x(x+1)}
  3. Subtract the numerators: 2x-(x+1)=2x-x-1=x-1
  4. Answer: \dfrac{x-1}{x(x+1)}
  5. (The minus sign applies to the whole of (x+1), so -1 not +1.)

Question 3Challenge6 marks

\dfrac{3x^2-10x-8}{x^2-6x+8}\div\dfrac{9x^2-4}{6x^2-4x}

(a)

Simplify the expression fully.

4 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

Hence solve

\dfrac{3x^2-10x-8}{x^2-6x+8}\div\dfrac{9x^2-4}{6x^2-4x}=x-3

2 marks

Give every value, separated by commas

Hint

Factorise all four expressions first, then turn the division into a multiplication by flipping the second fraction.

Worked solution

Part (a)

  1. 3x^2-10x-8=(3x+2)(x-4)
  2. x^2-6x+8=(x-2)(x-4)
  3. 9x^2-4=(3x-2)(3x+2) (difference of two squares)
  4. 6x^2-4x=2x(3x-2)
  5. Flip the second fraction and multiply: \dfrac{(3x+2)(x-4)}{(x-2)(x-4)}\times\dfrac{2x(3x-2)}{(3x-2)(3x+2)}
  6. Cancel (x-4), (3x+2) and (3x-2)
  7. Answer: \dfrac{2x}{x-2}

Part (b)

  1. Use part (a): \dfrac{2x}{x-2}=x-3
  2. Multiply both sides by (x-2): 2x=(x-3)(x-2)
  3. 2x=x^2-5x+6
  4. x^2-7x+6=0
  5. (x-1)(x-6)=0
  6. Neither value makes a denominator zero, so both are valid.
  7. Answer: x=1 or x=6

More on this topic: Algebraic Fractions worksheet with full solutions

All AQA Level 2 Further Maths practice questions

Unofficial practice questions written by Teach Me Maths. Not produced or endorsed by AQA.