Solving quadratic equations by completing the square | Maths

Starter quiz

Which of these is a solution for $(x - 3)^2 = 0$ ?

$x = -3$

$x = 3$ ✓

$x = 0$
Which of these is a solution for $x^2 = 0$ ?

$x = 0$ ✓

$x = 1$

There are no solutions.
Which of these is a solution for $(x - 4)(x + 2) = 0$ ?

$x = -2$ ✓

$x = 4$ ✓

$x = -8$

$x = 8$
The equation for this curve is $y=x^2 + 3x + 3$ . How many real solutions will it have?

0 ✓

1

2
Rearrange the following equation to make $m$ the subject: $3x - m = 9$

$m = 9 - 3x$

$m = 3x - 9$ ✓

$m = 3(x + 9)$
Rearrange the following equation to make $x$ the subject: $3x - m = 9$

$x = \frac{m + 9}{3}$ ✓

$x = \frac{m - 9}{3}$

$x = \frac{m + 3}{9}$

Exit quiz

Write $x^2 + 4x + 5$ in the form of $(x + p)^2 + q$

$(x + 2)^2 + 1$ ✓

$(x + 2)^2 - 1$

$(x - 2)^2 + 1$
Express $x^2 + 6x + 10$ as $(x + p)^2 + q$

$(x + 2)^2 - 1$

$(x + 3)^2 + 1$ ✓

$(x - 2)^2 + 1$
Express $x^2 + 8x + 14$ as $(x + p)^2 + q$

$(x - 4)^2 + 8$

$(x + 4)^2 - 2$ ✓

$(x + 4)^2 + 2$
Put $x^2 - 4x + 5$ in the form $(x + p)^2 + q$

$(x - 2)^2 + 1$ ✓

$(x - 2)^2 - 1$

$(x + 2)^2 + 1$
Solve by completing the square: $x^2 + 6x + 8 = 0$

$x = -4, x = -2$ ✓

$x = 4, x = 2$

$x = 4, x = -2$
Solve by completing the square: $x^2 + 6x + 5 = 0$

$x = 5, x = 1$

$x = 5, x = -1$

$x = -5, x = -1$ ✓

Worksheet

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Lesson Details

Key learning points

There are other methods to find the solutions of a quadratic equation.
One of these methods is called completing the square.
By representing this with a model, you can see why it has this name.
Completing the square is useful when the quadratic cannot be easily factorised.
The square root of a number can be positive or negative.

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.

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