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Year 10
Higher

Solving quadratic equations by completing the square

I can solve quadratic equations algebraically by completing the square.

Download all resources
Share activities with pupils
New
New
Year 10
Higher

Solving quadratic equations by completing the square

I can solve 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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These resources were created for remote use during the pandemic and are not designed for classroom teaching.

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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. By representing this with a model, you can see why it has this name.
  4. Completing the square is useful when the quadratic cannot be easily factorised.
  5. The square root of a number can be positive or negative.

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

That we subtract the constant squared when the constant in the bracket is positive and add the constant squared when the constant in the bracket is negative.

Squaring a negative value gives a positive value. Whether the constant in the bracket is positive or negative, the square of the constant must always be subtracted. This can be made clearer with algebra tiles or area models.


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

In the lesson, the algebra tiles are orientated on a pair of axes so that negative tiles are in the second and fourth quadrant. This is not essential for understanding but does help pupils see where partial products are positive and negative and makes the algebra tiles easier to place correctly.
Teacher tip

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 $$(x - 3)^2 = 0$$?
$$x = -3$$
Correct answer: $$x = 3$$
$$x = 0$$
Q2.
Which of these is a solution for $$x^2 = 0$$?
Correct answer: $$x = 0$$
$$x = 1$$
There are no solutions.
Q3.
Which of these is a solution for $$(x - 4)(x + 2) = 0$$?
Correct answer: $$x = -2$$
Correct answer: $$x = 4$$
$$x = -8$$
$$x = 8$$
Q4.
The equation for this curve is $$y=x^2 + 3x + 3$$. How many real solutions will it have?
An image in a quiz
Correct answer: 0
1
2
Q5.
Rearrange the following equation to make $$m$$ the subject: $$3x - m = 9$$
$$m = 9 - 3x$$
Correct answer: $$m = 3x - 9$$
$$m = 3(x + 9)$$
Q6.
Rearrange the following equation to make $$x$$ the subject: $$3x - m = 9$$
Correct answer: $$x = \frac{m + 9}{3}$$
$$x = \frac{m - 9}{3}$$
$$x = \frac{m + 3}{9}$$

Assessment exit quiz

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6 Questions

Q1.
Write $$x^2 + 4x + 5$$ in the form of $$(x + p)^2 + q$$
Correct answer: $$(x + 2)^2 + 1$$
$$(x + 2)^2 - 1$$
$$(x - 2)^2 + 1$$
Q2.
Express $$x^2 + 6x + 10$$ as $$(x + p)^2 + q$$
$$(x + 2)^2 - 1$$
Correct answer: $$(x + 3)^2 + 1$$
$$(x - 2)^2 + 1$$
Q3.
Express $$x^2 + 8x + 14$$ as $$(x + p)^2 + q$$
$$(x - 4)^2 + 8$$
Correct answer: $$(x + 4)^2 - 2$$
$$(x + 4)^2 + 2$$
Q4.
Put $$x^2 - 4x + 5$$ in the form $$(x + p)^2 + q$$
Correct answer: $$(x - 2)^2 + 1$$
$$(x - 2)^2 - 1$$
$$(x + 2)^2 + 1$$
Q5.
Solve by completing the square: $$x^2 + 6x + 8 = 0$$
Correct answer: $$x = -4, x = -2$$
$$x = 4, x = 2$$
$$x = 4, x = -2$$
Q6.
Solve by completing the square: $$x^2 + 6x + 5 = 0$$
$$x = 5, x = 1$$
$$x = 5, x = -1$$
Correct answer: $$x = -5, x = -1$$