Algebra and proofAdditional / Further Maths

Solving quadratics by completing the square

Make a squared bracket, then take both square roots.

  • Complete the square and isolate the squared bracket.
  • Include both the positive and negative square roots.
  • Rearrange to isolate . Keep surds exact.

Example 1

Take both square roots

Solve by completing the square.

Apply this to both sides.Take both square roots. Include positive and negative roots.Apply this to both square-root branches.

Example 2

Complete the square to find all real solutions.

Apply this to both sides.Take both square roots. Include positive and negative roots.Apply this to both square-root branches.

Your turn

Question 1

Find all real solutions.

Check answer
Apply this to both sides.A zero product has at least one zero factor. Set each factor equal to zero and solve each equation.
Question 2

Find all real solutions.

Check answer
Apply this to both sides.A zero product has at least one zero factor. Set each factor equal to zero and solve each equation.
Question 3

Find all real solutions.

Check answer
Apply this to both sides.A zero product has at least one zero factor. Set each factor equal to zero and solve each equation.
Question 4

Find all real solutions.

Check answer
Divide by the leading coefficient if needed, halve the coefficient of , then complete the square. Balance the added square on the other side.A zero product has at least one zero factor. Set each factor equal to zero and solve each equation.
Question 5

Find all real solutions.

Check answer
Apply this to both sides.A zero product has at least one zero factor. Set each factor equal to zero and solve each equation.
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