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Year 10
Higher

Solving complex quadratic equations by completing the square

I can solve more complex quadratic equations algebraically by completing the square.

Download all resources
Share activities with pupils
Year 10
Higher

Solving complex quadratic equations by completing the square

I can solve more complex quadratic equations algebraically by completing the square.

Download all resources
Share activities with pupils

These resources will be removed by end of Summer Term 2025.

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Lesson details

Key learning points

  1. There are other methods to find the solutions of a quadratic equation.
  2. One of these methods is called completing the square.
  3. Completing the square is useful when the quadratic cannot be easily factorised.
  4. Quadratic equations of the form $$ax^2 + bx + c = 0$$ (where $$a$$ ≠ 1) can be solved by completing the square.

Keywords

  • Completing the square - Completing the square is the process of rearranging an expression of the form ax^2 + bx + c into an equivalent expression of the form a(x + p)^2 + q

Common misconception

Pupils may believe that to write in completed square form they can divide through by the coefficient of x^2

This works for equations as equality can be maintained by dividing both sides of the equation. However if manipulating an expression, dividing through by a value changes the expression. Instead a value can be 'factored out' as necessary.


To help you plan your year 10 maths lesson on: Solving complex quadratic equations by completing the square, download all teaching resources for free and adapt to suit your pupils' needs...

Writing complex quadratics in completed square form can be a trickier concept than solving. Using algebra tiles to explore this can be beneficial for pupils of all abilities.
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Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Lesson video

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Prior knowledge starter quiz

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6 Questions

Q1.
Which of these is a solution for $$(x - 4)^2 = 0$$?
$$x = -4$$
Correct answer: $$x = 4$$
$$x = 0$$
Q2.
Which of these is a solution for $$x^2 - x = 0$$?
Correct answer: $$x = 0$$
$$x = -1$$
Correct answer: $$x = 1$$
Q3.
Which of these is a solution for $$(x + 4)(x + 3) = 0$$?
$$x = 4$$
$$x = 0$$
Correct answer: $$x = -4$$
Q4.
This image shows the equation $$(x+3)^2$$. How many solutions does $$(x+3)^2 = 0$$ have?
An image in a quiz
0
Correct answer: 1
2
Q5.
Rearrange the following equation to make $$w$$ the subject: $$5w - 3p = 10$$
$$w = \frac{3p + 10}{3}$$
Correct answer: $$w = \frac{3p + 10}{5}$$
$$w = \frac{3p - 10}{5}$$
Q6.
Rearrange the following equation to make $$p$$ the subject: $$5w - 3p = 10$$
Correct answer: $$p = \frac{5w - 10}{3}$$
$$p = \frac{5w - 10}{5}$$
$$p = \frac{5w + 10}{3}$$

Assessment exit quiz

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6 Questions

Q1.
Solve $$2x^2 + 12x + 10$$ by completing the square.
Correct answer: $$x = -5, -1$$
$$x = -6, -2$$
$$x = -4, 0$$
Q2.
Solve $$x^2 - 6x + 8$$ by completing the square.
$$x = 1, 3$$
$$x = 3, 5$$
Correct answer: $$x = 2, 4$$
Q3.
Solve $$3x^2 + 9x - 12$$ by completing the square
$$x = -3, 4$$
Correct answer: $$x = -4, 1$$
$$x = -5, 2$$
Q4.
Solve $$x^2 + 4x - 21$$ by completing the square.
$$x = -8, 4$$
$$x = -6, 2$$
Correct answer: $$x = -7, 3$$
Q5.
Solve $$4x^2 - 8x + 3$$ by completing the square.
Correct answer: $$x = 1/2, 3/2$$
$$x = -1/2, -3/2$$
$$x = 2/3, 4/3$$
Q6.
Solve $$x^2 + 10x + 16$$ by completing the square.
$$x = -9, -1$$
$$x = -7, -3$$
Correct answer: $$x = -8, -2$$