Algebraic fractions questions with answers

Questions

1. Question 1Cancel a common bracket

   Simplify $\dfrac{x^{2} - 9x + 18}{x - 3}$.

   Show the worked solution

   1. Step 1: Factorise every quadratic$\dfrac{x^{2} - 9x + 18}{x - 3} = \dfrac{(x - 3)(x - 6)}{(x - 3)}$
   2. Step 2: Cancel the common brackets (whole brackets only, never single terms)$= x - 6$
2. Question 2Quadratic over quadratic

   Simplify $\dfrac{x^{2} + 5x + 4}{x^{2} + 6x + 5}$.

   Show the worked solution

   1. Step 1: Factorise every quadratic$\dfrac{x^{2} + 5x + 4}{x^{2} + 6x + 5} = \dfrac{(x + 1)(x + 4)}{(x + 1)(x + 5)}$
   2. Step 2: Cancel the common brackets (whole brackets only, never single terms)$= \dfrac{x + 4}{x + 5}$
3. Question 3Add

   Express $\dfrac{3}{x - 5} + \dfrac{5}{x + 3}$ as a single fraction in its simplest form.

   Show the worked solution

   1. Step 1: Use a common denominator$\dfrac{3}{x - 5} + \dfrac{5}{x + 3} = \dfrac{3(x + 3) + 5(x - 5)}{(x - 5)(x + 3)}$
   2. Step 2: Expand the top (watch the sign in front of the second bracket)$= \dfrac{3x + 9 + (5x - 25)}{(x - 5)(x + 3)}$
   3. Step 3: Collect like terms$= \dfrac{8x - 16}{(x - 5)(x + 3)}$
4. Question 4Subtract

   Express $\dfrac{4}{x - 4} - \dfrac{1}{x + 2}$ as a single fraction in its simplest form.

   Show the worked solution

   1. Step 1: Use a common denominator$\dfrac{4}{x - 4} - \dfrac{1}{x + 2} = \dfrac{4(x + 2) - 1(x - 4)}{(x - 4)(x + 2)}$
   2. Step 2: Expand the top (watch the sign in front of the second bracket)$= \dfrac{4x + 8 - (x - 4)}{(x - 4)(x + 2)}$
   3. Step 3: Collect like terms$= \dfrac{3x + 12}{(x - 4)(x + 2)}$
5. Question 5Multiply

   Simplify $\dfrac{x^{2} - 2x - 8}{x^{2} - 5x + 6} \times \dfrac{x - 3}{x + 2}$.

   Show the worked solution

   1. Step 1: Write as one product$\dfrac{x^{2} - 2x - 8}{x^{2} - 5x + 6} \times \dfrac{x - 3}{x + 2}$$= \dfrac{x^{2} - 2x - 8}{x^{2} - 5x + 6} \times \dfrac{x - 3}{x + 2}$
   2. Step 2: Factorise every quadratic$= \dfrac{(x + 2)(x - 4)}{(x - 3)(x - 2)} \times \dfrac{x - 3}{x + 2}$
   3. Step 3: Cancel the common brackets (whole brackets only, never single terms)$= \dfrac{x - 4}{x - 2}$
6. Question 6Divide

   Simplify $\dfrac{x^{2} - 6x + 8}{x^{2} - 2x - 3} \div \dfrac{x - 4}{x - 3}$.

   Show the worked solution

   1. Step 1: Dividing: flip the second fraction and multiply$\dfrac{x^{2} - 6x + 8}{x^{2} - 2x - 3} \div \dfrac{x - 4}{x - 3}$$= \dfrac{x^{2} - 6x + 8}{x^{2} - 2x - 3} \times \dfrac{x - 3}{x - 4}$
   2. Step 2: Factorise every quadratic$= \dfrac{(x - 4)(x - 2)}{(x - 3)(x + 1)} \times \dfrac{x - 3}{x - 4}$
   3. Step 3: Cancel the common brackets (whole brackets only, never single terms)$= \dfrac{x - 2}{x + 1}$
7. Question 7Simplify: a² − b², a common factor or ax² + bx + c first

   Simplify $\dfrac{1 - x^{2}}{x^{2} - 4x + 3}$.

   Show the worked solution

   1. Step 1: Factorise the bottom$\dfrac{1 - x^{2}}{x^{2} - 4x + 3} = \dfrac{(1 - x)(x + 1)}{(x - 1)(x - 3)}$
   2. Step 2: 1 − x is −(x − 1): cancel it, leaving a minus$= -\dfrac{x + 1}{x - 3}$
8. Question 8A bracket the other way round: (k − x) = −(x − k)

   Express $\dfrac{7}{x - 5} + \dfrac{1}{5 - x}$ as a single fraction in its simplest form.

   Show the worked solution

   1. Step 1: Use 5 − x = −(x − 5): the sign in front of the second fraction changes$\dfrac{7}{x - 5} + \dfrac{1}{5 - x} = \dfrac{7}{x - 5} - \dfrac{1}{x - 5}$
   2. Step 2: Same denominator: add or subtract the tops$= \dfrac{6}{x - 5}$
9. Question 9Factorise a quadratic denominator first

   Express $\dfrac{4}{x^{2} + 7x + 10} + \dfrac{1}{x + 2}$ as a single fraction in its simplest form.

   Show the worked solution

   1. Step 1: Factorise the quadratic denominator$\dfrac{4}{x^{2} + 7x + 10} + \dfrac{1}{x + 2} = \dfrac{4}{(x + 2)(x + 5)} + \dfrac{1}{x + 2}$
   2. Step 2: Common denominator: multiply the second top by the missing bracket$= \dfrac{4 + 1(x + 5)}{(x + 2)(x + 5)}$
   3. Step 3: Expand the top and collect like terms$= \dfrac{x + 9}{(x + 2)(x + 5)}$
10. Question 10Multiply or divide: squares and differences of squares

    Simplify $\dfrac{x^{2} - 4}{x^{2} - 10x + 25} \div \dfrac{x + 2}{x - 5}$.

    Show the worked solution

    1. Step 1: Dividing: flip the second fraction and multiply$\dfrac{x^{2} - 4}{x^{2} - 10x + 25} \div \dfrac{x + 2}{x - 5} = \dfrac{x^{2} - 4}{x^{2} - 10x + 25} \times \dfrac{x - 5}{x + 2}$
    2. Step 2: Factorise the bottom of the first fraction: a perfect square$= \dfrac{(x + 2)(x - 2)}{(x - 5)^{2}} \times \dfrac{x - 5}{x + 2}$
    3. Step 3: Cancel the common brackets$= \dfrac{x - 2}{x - 5}$