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Square numbersExpanding quadratics

Solving equations by rearranging

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Here is everything you need to know about completing the square for GCSE maths (Edexcel, AQA and OCR). You’ll learn how to recognise a perfect square, complete the square on algebraic expressions, and tackle more difficult problems with the coefficient of ^{2} ≠ 1

You will also learn how to solve quadratic equations by completing the square, and how the completed square form links to graphs of quadratic equations.

Look out for completing the square worksheets and exam questions at the end.

A quadratic expression like ^{2} + 4x + 4**perfect square**.

This is because it factorises to give ^{2}

We can see this idea diagrammatically as follows:

Most quadratic expressions are not perfect squares, and cannot be written in this form as a single squared bracket. When we complete the square, we try to fit the expression to the closest possible perfect square, with a little bit added or subtracted to make things work.

Some expressions will have an ‘extra’ amount over from a perfect square, such as:

\[x^{2} + 4x + 7\]

So we would write this expression in completed square form as:

\[(x + 2)^{2} + 3\]

Some expressions will be ‘missing’ an amount to make a perfect square, such as:

\[x^{2} + 6x + 4\]

So we would write this expression in completed square form as:

\[(x + 3)^{2} – 5\]

While it’s easy to see this using diagrams for quadratic expressions with small coefficients, we need a better method for expressions with larger coefficients. You may have noticed already that we **divide the coefficient of **** by 2** in order to work out the nearest

Completing the square is a really useful method for solving quadratic equations; the quadratic formula for solving quadratic equations is based on it and can be derived by completing the square.

The completed square form of a quadratic expression is also really useful for identifying key points of quadratic functions, such as the **maximum **or **minimum **of a quadratic parabola (also called the vertex), without having to draw a graph. You can see this in the examples below.

You may sometimes see an expression in the form

In order to complete the square:

- Find the closest perfect square by dividing the coefficient of
x by2 . - Expand the perfect square expression.
- Compare the constant term in the perfect square to the original expression, and adjust as needed.

Get your free completing the square worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free completing the square worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONCompleting the square is part of the larger topic, quadratic equations. It may be useful to read about quadratic equations before exploring this topic and turning to its related lessons:

Complete the square for the expression

\[ x^{2} + 8x + 16\]

**Find the closest perfect square by dividing the coefficient of**.x by2

The coefficient of

The closest perfect square is:

\[(x + 4)^{2}\]

2 **Expand the perfect square expression**.

\[(x + 4)^{2} = x^{2} + 8x + 16\]

3 **Compare the constant term in the perfect square to the original expression, and adjust as needed**.

These match (because our example was a perfect square), so we don’t need to make any adjustment.

The answer in complete square form is

\[(x + 4)^{2}\]

**Graphically**

This graph shows the curve

\[y = x^{2} + 8x+ 16\]

The minimum value of

If we substitute

\[y = (-4 + 4)^{2} = 0^{2} = 0\]

So the coordinates of the vertex, which is a minimum point, are

Complete the square for the expression

\[x^{2} + 6x + 17\]

**Find the closest perfect square by dividing the coefficient of x by 2**.

The coefficient of

The closest perfect square is

\[(x + 3)^{2}\]

**Expand the perfect square expression**.

\[(x + 3)^{2} = x^{2} + 6x + 9\]

**Compare the constant term in the perfect square to the original expression, and adjust as needed**.

In order to make the constant term correct, we need to add

The answer in complete square form is

\[(x + 3)^{2} + 8\]

**Graphically**

This graph shows the curve

\[y = x^{2} + 6x + 17\]

The minimum value of

If we substitute

\[y = (-3 + 3)^{2} + 8 = 0^{2} + 8 = 8\]

So the coordinates of the vertex, which is a minimum point, are

As a little shortcut, the

Complete the square for the expression

\[x^{2} + 2x – 5\]

**Find the closest perfect square by dividing the coefficient of x by 2**.

The coefficient of

The closest perfect square is

\[(x +1)^{2}\]

**Expand the perfect square expression**.

\[(x + 1)^{2} = x^{2} + 2x + 1\]

**Compare the constant term in the perfect square to the original expression, and adjust as needed**.

In order to make the constant term correct, we need to subtract

The answer in complete square form is

\[(x + 1)^{2} – 6\]

**Graphically**

This graph shows the curve

\[y = x^{2} + 2x – 5\]

The minimum value of

If we substitute

\[y = (-1 + 1)^{2} – 6 = 0^{2} – 6 = -6\]

So the coordinates of the vertex, which is a minimum point, are

This is really straightforward – just remember that your perfect square bracket will need subtraction rather than addition in the middle.

Complete the square for the expression

\[x^{2} – 10x + 17\]

**Find the closest perfect square by dividing the coefficient of x by 2**.

The coefficient of

The closest perfect square is

\[(x – 5)^{2}\]

**Expand the perfect square expression**.

\[(x – 5)^{2} = x^{2} – 10x + 25\]

**Compare the constant term in the perfect square to the original expression, and adjust as needed**.

In order to make the constant term correct, we need to subtract

The answer in complete square form is

\[(x – 5)^{2} – 8\]

**Graphically**

This graph shows the curve

\[y = x^{2} – 10x + 17\]

The minimum value of

If we substitute

\[y = (5 – 5)^{2} – 8 = 0^{2} – 8 = -8\]

So the coordinates of the vertex, which is a minimum point, are

Complete the square for the expression

\[x^{2} + 3x + 4\]

**Find the closest perfect square by dividing the coefficient of x by 2**.

The coefficient of

\[\frac{3}{2}\]

It can be tempting to use decimals, but fractions are much easier, particularly as completing the square is more likely to be examined on a non-calculator paper at GCSE.

The closest perfect square is

\[\left(x + \frac{3}{2}\right)^{2}\]

**Expand the perfect square expression**.

\[\left(x + \frac{3}{2}\right)^{2} = x^{2} + 3 x + \frac{9}{4}\]

**Compare the constant term in the perfect square to the original expression, and adjust as needed**.

It can be useful to think of

\[\frac{16}{4}\]

this makes it easier to work out the adjustment.

In order to make the constant term correct, we need to add

\[\frac{7}{4}\]

because

\[\frac{9}{4} + \frac{7}{4} = \frac{16}{4} = 4\]

The answer in complete square form is

\[\left(x+\frac{3}{2}\right)^{2}+\frac{7}{4}\]

**Graphically**

This graph shows the curve

\[y=x^{2}+3x+4\]

The minimum value of

\[x=\frac{-3}{2}\]

If we substitute

\[x=\frac{-3}{2}\]

we get:

\[y=\left(\frac{-3}{2}+\frac{3}{2}\right)^{2}+\frac{7}{4}=0^{2}+\frac{7}{4}=\frac{7}{4}\]

So the coordinates of the vertex, which is a minimum point, are

\[\left(\frac{-3}{2}, \frac{7}{4}\right)\]

If the coefficient of ^{2 }^{2 }^{2}

- Factorise.
- Complete the square on the expression inside the brackets: find the closest perfect square by dividing the coefficient of
x by2 . - Expand the perfect square expression.
- Compare the constant term in the perfect square to the original expression, and adjust as needed.
- Multiply out the factorised value.

Complete the square for the expression

\[2x^{2} + 8x – 10\]

** Factorise**.

We take out the common factor of

\[2\left(x^{2} + 4 x-5\right)\]

then complete the square on the expression inside the bracket.

**Find the closest perfect square by dividing the coefficient of x by 2.**

The coefficient of

The closest perfect square is

\[(x+2)^{2}\]

**Expand the perfect square expression**.

\[(x+2)^{2}=x^{2}+4x+4\]

**Compare the constant term in the perfect square to the original expression, and adjust as needed**.

In order to make the constant term correct, we need to subtract

So the expression in complete square form is

\[(x+2)^{2}-9\]

**Multiply out the factorised value.**

Don’t forget to deal with the factor of

\[2\left(x^{2}+4 x-5\right)\]

in complete square form is:

\[2\left((x+2)^{2}-9\right)\]

Finally, expand out to give a final answer:

\[2(x+2)^{2}-18\]

**Graphically**

This graph shows the curve

\[y=2x^{2}+8x-10\]

The minimum value of

If we substitute

\[y=2(x+2)^{2}-18=0^{2}-18=-18\]

So the coordinates of the vertex, which is a minimum point, are

Note: we can also complete the square for expressions with a negative ^{2}

For example,

\[-x^{2}+5x-3\]

would be written as:

\[-\left(x^{2}-5 x+3\right)\]

and then completed as in Example 5.

Some quadratic equations can be solved by factorising, but most require either completing the square or the quadratic formula – in fact, the quadratic formula is derived from completing the square (see worksheet).

In order to solve quadratic equations using complete the square:

- Make sure the equation is rearranged so that the right hand side equals
0 (if necessary). - Complete the square on the left hand side.
- Rearrange and solve the resulting equation for
x .

Solve the quadratic equation

\[x^{2}+6x+3=0\]

**Make sure the equation is rearranged so that the right hand side equals 0 (if necessary)**.

RHS already equals

**Complete the square on the left hand side**.

\[x^{2}+6x+3=(x+3)^{2}-6\]

**Rearrange and solve the resulting equation for x**.

\[\begin{aligned}
(x+3)^{2}-6&=0 \\\\
(x+3)^{2}&=6 \\\\
x+3&=\pm \sqrt{6} \\\\
x&=-3 \pm \sqrt{6}
\end{aligned}
\]

We have two solutions to the equation:

\[x=-3+\sqrt{6} \\\\
x=-3-\sqrt{6}
\]

It is really important to remember the

In most cases, your answer should be left in surd form (with the square root sign in), not converted to a decimal.

Solve the quadratic equation

\[2x^{2}+10x=-5-2x\]

**Make sure the equation is rearranged so that the right hand side equals 0 (if necessary)**.

Rearrange by adding

\[2x^{2}+12x+5=0\]

**Complete the square on the left hand side**.

Remember to take out a common factor of

\[\begin{aligned}
2 x^{2}+12 x+5 &=2\left[x^{2}+6 x+\frac{5}{2}\right] \\\\
&=2\left[(x+3)^{2}-\frac{13}{2}\right] \\\\
&=2(x+3)^{2}-13
\end{aligned}
\]

**Rearrange and solve the resulting equation for x**.

\begin{aligned}
2(x+3)^{2}-13&=0 \\\\
2(x+3)^{2}&=13 \\\\
(x+3)^{2}&=\frac{13}{2} \\\\
x+3&=\pm \sqrt{\frac{13}{2}} \\\\
x&=-3 \pm \sqrt{\frac{13}{2}}
\end{aligned}

We have two solutions to the equation:

\[x=-3+\sqrt{\frac{13}{2}} \\\\
x=-3-\sqrt{\frac{13}{2}}
\]

**Incorrect sign in the middle of the perfect square bracket**

The sign will always be the same sign as the coefficient of

**Errors with fraction arithmetic**

In particular, remember to square the numerator and denominator when squaring a fraction.

**Going the wrong way when working out the adjustment**

Remember to always work out how you get ** from** the perfect square

**Forgetting the ± sign when using the completing the square method to solve quadratics**

Check carefully to make sure you haven’t lost one of your solutions!

1. Complete the square:

x^{2} + 12x + 36

(x+12)^{2}

(x+6)^{2}

(x-6)^{2}

(x+6)^{2} +36

Half the coefficient of x is 6 so it is (x+6)^{2} .

Expanding (x+6)^{2}. gives us x^{2}+12x+36 so we do not need to add or subtract anything.

2. Complete the square:

x^{2}+6x+2

(x+3)^{2}-7

(x+3)^{2}+7

(x-3)^{2}-7

(x+1)^{2}+1

Half the coefficient of x is 3 so it is (x+3)^{2} .

Expanding (x+3)^{2} gives us x^{2}+6x+9 so we need to subtract 7 which gives us (x+3)^{2} -7 .

3. Complete the square:

x^{2}-5x+12

=(x-5)^{2}-13

=(x+\frac{5}{2})^{2}+\frac{23}{4}

=(x-\frac{5}{2})^{2}+\frac{73}{4}

=(x-\frac{5}{2})^{2}+\frac{23}{4}

Half the coefficient of x is \frac{5}{2} so it is (x-\frac{5}{2})^{2} .

Expanding (x-\frac{5}{2})^{2} gives us x^{2}-5x+\frac{25}{4} .

We need +12 , which is the same as + \frac{48}{4} so we need to add \frac{23}{4} .

Therefore the answer is (x-\frac{5}{2})^{2} + \frac{23}{4} .

4. Complete the square:

-2x^{2}-4x+14

2(-x-2)^{2}+20

-2(x+1)^{2}+16

-(2x-2)^{2}+10

-2(x-1)^{2}-26

We must first take a factor of -2 out to get -2(x^{2}+2x-7) .

We then need to complete the square for x^{2}+2x-7 .

This gives us (x+1)^{2}-8 .

So the final answer is -2[(x+1)^{2}-8]=-2(x+1)^{2}+16 .

5. Find the coordinates of the minimum point of the function:

y=x^{2}-6x+1

(6, -35)

(-3, -8)

(3, -8)

(-6, 1)

We need to start by completing the square. This gives us y=(x-3)^{2}-8 .

We can then see that the minimum point is when (x-3) = 0 so x=3 .

Substituting this in gives us y=0^{2}-8=-8. .

Therefore the minimum point is (3, -8) .

6. By first completing the square, find the solutions to the equation:

x^{2}+8x+3=0

x=-4 \pm \sqrt{13}

x=-4 \pm \sqrt{3}

x=4 \pm \sqrt{13}

x=-4 + \sqrt{13}

Completing the square gives us (x+4)^{2}-13 .

Therefore we need to solve:

\begin{aligned} (x+4)^{2}-13&=0\\\\ (x+4)^{2}&=13\\\\ x+4&=\pm \sqrt{13}\\\\ x&= -4 \pm \sqrt{13} \end{aligned}1. x^{2}+12x+19 can be written in the form

(x+a)^{2}+b where a and b are integers.

Work out the values of a and b .

**(2 marks)**

Show answer

a = 6

**(1)**

**(1)**

2. (a) x^{2}-6x+7 can be written in the form

(x+a)^{2}+b where a and b are integers.

Work out the values of a and b .

(b) Hence, or otherwise, solve the equation

x^{2}-6x+7=0Give your answers in surd form.

**(4 marks)**

Show answer

(a)

a = −3**(1)**

**(1)**

(b)

x = 3 + \sqrt{2}**(1)**

**(1)**

3. (a) 3x^{2}+12x-18 can be written in the form

a(x+b)^{2}+c where a and b are integers.

Work out the values of a, b and c .

(b) Hence, or otherwise, find the coordinates of the turning point of the graph of y=3x^{2}+12x-18

**(4 marks)**

Show answer

(a)

a = 3**(1)**

**(1)**

**(1)**

(b)

(−2, −30)**(1)**

You have now learned how to:

- Complete the square for a quadratic expression
- Solve quadratic equations by completing the square
- Identify turning points by completing the square

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