2.2 Completing the Square

What you'll learn

How to rewrite a quadratic in completed-square form.
How completed square form shows the minimum point of a parabola.
How to prove a quadratic is always positive.
How to solve quadratic equations, including ones with constants such as $p$ or $k$ .

1. The key prerequisite: expanding a square

Before you can complete the square, you need to be comfortable expanding brackets such as $x+3)^2$ .

Definition

Perfect square

A perfect square is an expression made by squaring a bracket, such as $x+4)^2$ or $x-7)^2$ .

The important pattern is:

$x+a)^2 = x^2 + 2ax + a^2$

So the middle coefficient is always twice the number inside the bracket.

Example

Expanding a squared bracket

Expand $x-5)^2$ .

Write the square as two identical brackets:

$x-5)^2=(x-5)(x-5)$
Multiply out carefully:

$x^2-5x-5x+25$
Collect like terms:

$x-5)^2=x^2-10x+25$

Key Idea

The halfway idea

To complete the square for $x^2+bx+c$ , halve the coefficient of $x$ first. That number goes inside the bracket.

2. Completing the square when the coefficient of $x^2$ is 1

A quadratic like $x^2+10x+7$ is close to a perfect square. The square with the same $x^2$ and $x$ terms is $x+5)^2$ , because 5 is half of 10.

Definition

Completed-square form

For a quadratic with coefficient 1 on $x^2$ , completed-square form means writing it as

$x+a)^2+b$

where $a$ and $b$ are constants.

Example

Writing a quadratic in completed-square form

Express $x^2+10x+7$ in the form $x+a)^2+b$ .

Halve the coefficient of $x$ :

$102=5\frac{10}{2}=5$
Start with the matching square:

$x+5)^2=x^2+10x+25$
Compare this with the original expression. The square has 25, but the original has 7, so subtract 18:

$x^2+10x+7=(x+5)^2-18$

Common Mistake

Forgetting to correct the constant

If you write $x^2+10x+7=(x+5)^2+7$ , you have added an extra 25 by accident. Always expand your answer mentally to check it returns to the original quadratic.

When the coefficient of $x$ is odd

If the coefficient of $x$ is odd, you will usually get fractions. That is completely normal at AS level.

Example

Completing the square with fractions

Express $x^2-7x+4$ in completed-square form.

Halve the coefficient of $x$ :

$−72=−72\frac{-7}{2}=-\frac{7}{2}$
Use this inside the bracket:

$(x−72)2\left(x-\frac{7}{2}\right)^2$
Expand the square so you can see the extra constant:

$(x−72)2=x2−7x+494\left(x-\frac{7}{2}\right)^2=x^2-7x+\frac{49}{4}$
Adjust the constant from $494\frac{49}{4}$ down to 4:

$x2−7x+4=(x−72)2−334x^2-7x+4=\left(x-\frac{7}{2}\right)^2-\frac{33}{4}$

Tip

Quick check

In $x+a)^2+b$ , the coefficient of $x$ will be $2 a$ . So if your original quadratic has $- 7 x$ , the bracket must contain $−72-\frac{7}{2}$ .

3. When the coefficient of $x^2$ is not 1

A non-monic quadratic is a quadratic where the coefficient of $x^2$ is not 1, such as $2x^2+8x+1$ .

Definition

Monic and non-monic

A quadratic is monic if the coefficient of $x^2$ is 1. It is non-monic if the coefficient of $x^2$ is any other non-zero number.

For non-monic quadratics, first factor out the coefficient of $x^2$ from the $x^2$ and $x$ terms.

A step diagram showing the structure of completing the square for a non-monic quadratic.

Example

Completing the square when the leading coefficient is not 1

Write $3x^2-12x+5$ in the form $A(x+B)^2+C$ .

Factor 3 out of the $x^2$ and $x$ terms:

$3x^2-12x+5=3(x^2-4x)+5$
Complete the square inside the bracket:

$x^2-4x=(x-2)^2-4$
Substitute this back in:

$3(x2−4x)+5=3((x−2)2−4)+53(x^2-4x)+5=3\left((x-2)^2-4\right)+5$
Expand the outside 3 through the constant part and simplify:

$3(x-2)^2-12+5=3(x-2)^2-7$
Read off the constants:

$A=3,B=−2,C=−7A=3,\quad B=-2,\quad C=-7$

4. Using completed-square form to find a minimum point

The expression $x+2)^2$ can never be negative, because squaring any real number gives a value greater than or equal to zero.

So for a curve like $y=(x+2)^2-3$ , the smallest possible value of the square is zero. That makes the minimum point $(- 2, - 3)$ .

Graph of the parabola y = (x + 2)^2 - 3, labelled with vertex (-2, -3) and axis of symmetry x = -2

Definition

Minimum point

The minimum point of a parabola is its lowest point. For $y=(x-a)^2+b$ , the minimum point is $(a, b)$ .

Example

Finding the minimum point

Find the minimum point of the curve $y=x^2-8x+11$ .

Complete the square:

$x^2-8x+11=(x-4)^2-5$
Use the fact that $(x−4)2≥0(x-4)^2 \ge 0$ for all real $x$ .
The smallest value occurs when the square equals zero:

$x - 4 = 0$
Therefore $x = 4$ , and the minimum value of $y$ is -5.
The minimum point is:

$(4, - 5)$

Example

Finding constants from a given minimum point

The curve $y=x^2+ax+b$ has minimum point $(3, - 4)$ . Find $a$ and $b$ .

Use the vertex form directly. Since the minimum point is $(3, - 4)$ , the curve can be written as:

$y=(x-3)^2-4$
Expand the bracket:

$x-3)^2-4=x^2-6x+9-4$
Simplify:

$y=x^2-6x+5$
Compare with $y=x^2+ax+b$ :

$a=−6,b=5a=-6,\quad b=5$

5. Proving a quadratic is always positive

Completed-square form is very powerful for inequalities because a square is always at least zero.

Key Idea

Positive for all real values

If you can write a quadratic as $x+a)^2+c$ where $c > 0$ , then the whole expression is always positive.

Example

Showing a quadratic is always positive

Show that $x^2-10x+28>0$ for all real values of $x$ .

Complete the square:

$x^2-10x+28=(x-5)^2+3$
Use the square fact:

$(x−5)2≥0(x-5)^2 \ge 0$
Add 3 to both sides mentally, so the smallest possible value is 3.
Therefore:

$x^2-10x+28>0$

Example

Proving one expression is bigger than another

Prove that $x^2+3x+9>x+1$ for all real $x$ .

Move everything to the left-hand side:

$x^2+3x+9-(x+1)>0$
Simplify:

$x^2+2x+8>0$
Complete the square:

$x^2+2x+8=(x+1)^2+7$
Since $(x+1)2≥0(x+1)^2 \ge 0$ , the smallest possible value of $x+1)^2+7$ is 7.
Therefore $x^2+3x+9>x+1$ for all real $x$ .

6. Solving equations by completing the square

A root of an equation is a value of $x$ that makes the equation true. For a quadratic equation, the roots are where the graph crosses the $x$ -axis.

Example

Solving a quadratic with a constant parameter

By completing the square, find the roots of $x^2+px+5=0$ in terms of $p$ .

Complete the square:

$x2+px+5=(x+p2)2−p24+5x^2+px+5=\left(x+\frac{p}{2}\right)^2-\frac{p^2}{4}+5$
Set the completed-square form equal to zero:

$(x+p2)2−p24+5=0\left(x+\frac{p}{2}\right)^2-\frac{p^2}{4}+5=0$
Rearrange to isolate the square:

$(x+p2)2=p24−5\left(x+\frac{p}{2}\right)^2=\frac{p^2}{4}-5$
Take square roots of both sides:

$x+p2=±p24−5x+\frac{p}{2}=\pm \sqrt{\frac{p^2}{4}-5}$
Write the roots neatly:

$x=−p2±12p2−20x=-\frac{p}{2}\pm \frac{1}{2}\sqrt{p^2-20}$

Common Mistake

No real roots

When solving by square rooting, the expression under the square root must be non-negative if you want real roots. If it is negative, the quadratic has no real roots.

Example

Finding when there are no real roots

Find the values of $p$ for which $x^2+px+5=0$ has no real roots.

$The parameter number line shows the interval <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>2</mn><msqrt><mn>5</mn></msqrt><mo><</mo><mi>p</mi><mo><</mo><mn>2</mn><msqrt><mn>5</mn></msqrt></mrow><annotation encoding="application/x-tex">-2\sqrt5<p<2\sqrt5</annotation></semantics></math>−25<svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/></svg><p<25<svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/></svg> where the quadratic has no real roots.$

From the completed-square solution, the roots involve:

$p2−20\sqrt{p^2-20}$
For no real roots, the expression under the square root must be negative:

$p^2-20<0$
Rearrange:

$p^2<20$
Take square roots carefully:

$−25<p<25-2\sqrt{5}<p<2\sqrt{5}$

Exam technique

In the exam

Always show the completed-square line clearly; it often earns method marks even if the final answer goes wrong.
For minimum points, remember that $x-a)^2+b$ has minimum point $(a, b)$ , not $(- a, b)$ .
For “no real roots” questions, focus on the expression under the square root and set it less than zero.

Self review

Check yourself

Can you complete the square for $x^2+12x-7$ without expanding fully?
What is the minimum point of $y=(x+6)^2-11$ ?
For $x^2+kx+10=0$ , what expression would appear under the square root when solving by completing the square?

2.2 Completing the Square

What you'll learn

How to rewrite a quadratic in completed-square form.
How completed square form shows the minimum point of a parabola.
How to prove a quadratic is always positive.
How to solve quadratic equations, including ones with constants such as $p$ or $k$ .

1. The key prerequisite: expanding a square

Before you can complete the square, you need to be comfortable expanding brackets such as $x+3)^2$ .

Definition

Perfect square

A perfect square is an expression made by squaring a bracket, such as $x+4)^2$ or $x-7)^2$ .

The important pattern is:

$x+a)^2 = x^2 + 2ax + a^2$

So the middle coefficient is always twice the number inside the bracket.

Example

Expanding a squared bracket

Expand $x-5)^2$ .

Write the square as two identical brackets:

$x-5)^2=(x-5)(x-5)$
Multiply out carefully:

$x^2-5x-5x+25$
Collect like terms:

$x-5)^2=x^2-10x+25$

Key Idea

The halfway idea

To complete the square for $x^2+bx+c$ , halve the coefficient of $x$ first. That number goes inside the bracket.

2. Completing the square when the coefficient of $x^2$ is 1

A quadratic like $x^2+10x+7$ is close to a perfect square. The square with the same $x^2$ and $x$ terms is $x+5)^2$ , because 5 is half of 10.

Definition

Completed-square form

For a quadratic with coefficient 1 on $x^2$ , completed-square form means writing it as

$x+a)^2+b$

where $a$ and $b$ are constants.

Example

Writing a quadratic in completed-square form

Express $x^2+10x+7$ in the form $x+a)^2+b$ .

Halve the coefficient of $x$ :

$102=5\frac{10}{2}=5$
Start with the matching square:

$x+5)^2=x^2+10x+25$
Compare this with the original expression. The square has 25, but the original has 7, so subtract 18:

$x^2+10x+7=(x+5)^2-18$

Common Mistake

Forgetting to correct the constant

If you write $x^2+10x+7=(x+5)^2+7$ , you have added an extra 25 by accident. Always expand your answer mentally to check it returns to the original quadratic.

When the coefficient of $x$ is odd

If the coefficient of $x$ is odd, you will usually get fractions. That is completely normal at AS level.

Example

Completing the square with fractions

Express $x^2-7x+4$ in completed-square form.

Halve the coefficient of $x$ :

$−72=−72\frac{-7}{2}=-\frac{7}{2}$
Use this inside the bracket:

$(x−72)2\left(x-\frac{7}{2}\right)^2$
Expand the square so you can see the extra constant:

$(x−72)2=x2−7x+494\left(x-\frac{7}{2}\right)^2=x^2-7x+\frac{49}{4}$
Adjust the constant from $494\frac{49}{4}$ down to 4:

$x2−7x+4=(x−72)2−334x^2-7x+4=\left(x-\frac{7}{2}\right)^2-\frac{33}{4}$

Tip

Quick check

In $x+a)^2+b$ , the coefficient of $x$ will be $2 a$ . So if your original quadratic has $- 7 x$ , the bracket must contain $−72-\frac{7}{2}$ .

3. When the coefficient of $x^2$ is not 1

A non-monic quadratic is a quadratic where the coefficient of $x^2$ is not 1, such as $2x^2+8x+1$ .

Definition

Monic and non-monic

A quadratic is monic if the coefficient of $x^2$ is 1. It is non-monic if the coefficient of $x^2$ is any other non-zero number.

For non-monic quadratics, first factor out the coefficient of $x^2$ from the $x^2$ and $x$ terms.

A step diagram showing the structure of completing the square for a non-monic quadratic.

Example

Completing the square when the leading coefficient is not 1

Write $3x^2-12x+5$ in the form $A(x+B)^2+C$ .

Factor 3 out of the $x^2$ and $x$ terms:

$3x^2-12x+5=3(x^2-4x)+5$
Complete the square inside the bracket:

$x^2-4x=(x-2)^2-4$
Substitute this back in:

$3(x2−4x)+5=3((x−2)2−4)+53(x^2-4x)+5=3\left((x-2)^2-4\right)+5$
Expand the outside 3 through the constant part and simplify:

$3(x-2)^2-12+5=3(x-2)^2-7$
Read off the constants:

$A=3,B=−2,C=−7A=3,\quad B=-2,\quad C=-7$

4. Using completed-square form to find a minimum point

The expression $x+2)^2$ can never be negative, because squaring any real number gives a value greater than or equal to zero.

So for a curve like $y=(x+2)^2-3$ , the smallest possible value of the square is zero. That makes the minimum point $(- 2, - 3)$ .

Graph of the parabola y = (x + 2)^2 - 3, labelled with vertex (-2, -3) and axis of symmetry x = -2

Definition

Minimum point

The minimum point of a parabola is its lowest point. For $y=(x-a)^2+b$ , the minimum point is $(a, b)$ .

Example

Finding the minimum point

Find the minimum point of the curve $y=x^2-8x+11$ .

Complete the square:

$x^2-8x+11=(x-4)^2-5$
Use the fact that $(x−4)2≥0(x-4)^2 \ge 0$ for all real $x$ .
The smallest value occurs when the square equals zero:

$x - 4 = 0$
Therefore $x = 4$ , and the minimum value of $y$ is -5.
The minimum point is:

$(4, - 5)$

Example

Finding constants from a given minimum point

The curve $y=x^2+ax+b$ has minimum point $(3, - 4)$ . Find $a$ and $b$ .

Use the vertex form directly. Since the minimum point is $(3, - 4)$ , the curve can be written as:

$y=(x-3)^2-4$
Expand the bracket:

$x-3)^2-4=x^2-6x+9-4$
Simplify:

$y=x^2-6x+5$
Compare with $y=x^2+ax+b$ :

$a=−6,b=5a=-6,\quad b=5$

5. Proving a quadratic is always positive

Completed-square form is very powerful for inequalities because a square is always at least zero.

Key Idea

Positive for all real values

If you can write a quadratic as $x+a)^2+c$ where $c > 0$ , then the whole expression is always positive.

Example

Showing a quadratic is always positive

Show that $x^2-10x+28>0$ for all real values of $x$ .

Complete the square:

$x^2-10x+28=(x-5)^2+3$
Use the square fact:

$(x−5)2≥0(x-5)^2 \ge 0$
Add 3 to both sides mentally, so the smallest possible value is 3.
Therefore:

$x^2-10x+28>0$

Example

Proving one expression is bigger than another

Prove that $x^2+3x+9>x+1$ for all real $x$ .

Move everything to the left-hand side:

$x^2+3x+9-(x+1)>0$
Simplify:

$x^2+2x+8>0$
Complete the square:

$x^2+2x+8=(x+1)^2+7$
Since $(x+1)2≥0(x+1)^2 \ge 0$ , the smallest possible value of $x+1)^2+7$ is 7.
Therefore $x^2+3x+9>x+1$ for all real $x$ .

6. Solving equations by completing the square

A root of an equation is a value of $x$ that makes the equation true. For a quadratic equation, the roots are where the graph crosses the $x$ -axis.

Example

Solving a quadratic with a constant parameter

By completing the square, find the roots of $x^2+px+5=0$ in terms of $p$ .

Complete the square:

$x2+px+5=(x+p2)2−p24+5x^2+px+5=\left(x+\frac{p}{2}\right)^2-\frac{p^2}{4}+5$
Set the completed-square form equal to zero:

$(x+p2)2−p24+5=0\left(x+\frac{p}{2}\right)^2-\frac{p^2}{4}+5=0$
Rearrange to isolate the square:

$(x+p2)2=p24−5\left(x+\frac{p}{2}\right)^2=\frac{p^2}{4}-5$
Take square roots of both sides:

$x+p2=±p24−5x+\frac{p}{2}=\pm \sqrt{\frac{p^2}{4}-5}$
Write the roots neatly:

$x=−p2±12p2−20x=-\frac{p}{2}\pm \frac{1}{2}\sqrt{p^2-20}$

Common Mistake

No real roots

When solving by square rooting, the expression under the square root must be non-negative if you want real roots. If it is negative, the quadratic has no real roots.

Example

Finding when there are no real roots

Find the values of $p$ for which $x^2+px+5=0$ has no real roots.

From the completed-square solution, the roots involve:

$p2−20\sqrt{p^2-20}$
For no real roots, the expression under the square root must be negative:

$p^2-20<0$
Rearrange:

$p^2<20$
Take square roots carefully:

$−25<p<25-2\sqrt{5}<p<2\sqrt{5}$

Exam technique

In the exam

Always show the completed-square line clearly; it often earns method marks even if the final answer goes wrong.
For minimum points, remember that $x-a)^2+b$ has minimum point $(a, b)$ , not $(- a, b)$ .
For “no real roots” questions, focus on the expression under the square root and set it less than zero.

Self review

Check yourself

Can you complete the square for $x^2+12x-7$ without expanding fully?
What is the minimum point of $y=(x+6)^2-11$ ?
For $x^2+kx+10=0$ , what expression would appear under the square root when solving by completing the square?

2.2 Completing the Square

What you'll learn

1. The key prerequisite: expanding a square

2. Completing the square when the coefficient of $x^2$ is 1

When the coefficient of $x$ is odd

3. When the coefficient of $x^2$ is not 1

4. Using completed-square form to find a minimum point

5. Proving a quadratic is always positive

6. Solving equations by completing the square

Recap questions

You've reached the end

2.2 Completing the Square

What you'll learn

1. The key prerequisite: expanding a square

2. Completing the square when the coefficient of $x^2$ is 1

When the coefficient of $x$ is odd

3. When the coefficient of $x^2$ is not 1

4. Using completed-square form to find a minimum point

5. Proving a quadratic is always positive

6. Solving equations by completing the square

Recap questions

You've reached the end

2.2 Completing the Square

What you'll learn

1. The key prerequisite: expanding a square

2. Completing the square when the coefficient of x2x^2x2 is 1

When the coefficient of xxx is odd

3. When the coefficient of x2x^2x2 is not 1

4. Using completed-square form to find a minimum point

5. Proving a quadratic is always positive

6. Solving equations by completing the square

Recap questions

You've reached the end

2.2 Completing the Square Revision Guide

2.2 Completing the Square

What you'll learn

1. The key prerequisite: expanding a square

2. Completing the square when the coefficient of x2x^2x2 is 1

When the coefficient of xxx is odd

3. When the coefficient of x2x^2x2 is not 1

4. Using completed-square form to find a minimum point

5. Proving a quadratic is always positive

6. Solving equations by completing the square

Recap questions

You've reached the end

2.2 Completing the Square Revision Guide

2. Completing the square when the coefficient of $x^2$ is 1

When the coefficient of $x$ is odd

3. When the coefficient of $x^2$ is not 1

2. Completing the square when the coefficient of $x^2$ is 1

When the coefficient of $x$ is odd

3. When the coefficient of $x^2$ is not 1