Solving Quadratic Equations: Completing the Square

The Power of Completing the Square

Imagine you are faced with a quadratic equation that simply will not factorise. Completing the square is a powerful algebraic method used to rewrite a quadratic expression from the standard form $ax^2 + bx + c$ into the form $a(x + p)^2 + q$ . This process forces the creation of a perfect square trinomial, which is a quadratic expression that can be factorised into a single squared bracket, such as $x^2 + 6x + 9 = (x + 3)^2$ .

The "Half and Square" Rule

To complete the square for a quadratic where $a = 1$ (e.g., $x^2 + bx + c$ ), you focus entirely on the $x$ coefficient (the number $b$ in front of the $x$ ). You halve this coefficient to find the value inside your bracket, and then subtract its square on the outside to keep the equation balanced.

The core formula for completing the square is:

(x + \frac{b}{2})^2 - (\frac{b}{2})^2 + c

If the coefficient is negative, the bracket will be $(x - p)^2$ . Note that the square of a negative number is positive, but you must still subtract it outside the bracket.
If the coefficient is odd, leave the halved value as a fraction (e.g., $\frac{3}{2}$ ) rather than a decimal, as it makes squaring much easier (e.g., $(\frac{3}{2})^2 = \frac{9}{4}$ ).

Solving Quadratic Equations ( $a = 1$ )

Before you can solve, the equation must be set to zero. This may require quadratic rearrangement to get all terms onto one side in the form $x^2 + bx + c = 0$ . Once the square is completed, isolate the bracket and take the square root of both sides.

When square-rooting, you must include the $\pm$ symbol to account for both the positive and negative root.
Edexcel Higher tier papers frequently specify that answers must be left in exact form or surd form (e.g., $x = a \pm b\sqrt{c}$ ) rather than decimals.
Often, you will need to leave your answer in simplified surd form, ensuring the number under the root has no hidden square factors.

Worked Example 1: Solving and Surd Simplification

Solve $2x^2 - 11x - 25 = x^2 - x$ by completing the square. Give your answer in simplified surd form.

Step 1: Rearrange the equation into the standard form $ax^2 + bx + c = 0$ .

$x^2 - 10x - 25 = 0$

Step 2: Halve the $x$ -coefficient ( $-10 \div 2 = -5$ ) to create the perfect square, then subtract its square.

$(x - 5)^2 - (-5)^2 - 25 = 0$

Step 3: Simplify the constant terms.

$(x - 5)^2 - 25 - 25 = 0$
$(x - 5)^2 - 50 = 0$

Step 4: Rearrange to isolate the squared bracket.

$(x - 5)^2 = 50$

Step 5: Square root both sides, remembering the $\pm$ symbol.

$x - 5 = \pm\sqrt{50}$

Step 6: Simplify the surd by extracting the largest square factor of $50$ ( $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$ ).

$x - 5 = \pm 5\sqrt{2}$

Step 7: Solve for $x$ .

$x = 5 \pm 5\sqrt{2}$

Handling Non-Unitary Coefficients

A non-unitary coefficient occurs when the number in front of the $x^2$ term is not $1$ (for example, $2x^2$ or $-x^2$ ). If you are solving an equation like $ax^2 + bx + c = 0$ , the easiest approach is to divide every single term by $a$ first to simplify the process back to the $a = 1$ method.

Worked Example 2: Solving with $a > 1$

Solve $2x^2 + 12x + 4 = 0$ by completing the square. Leave your answer in exact form.

Step 1: Divide the entire equation by the non-unitary coefficient ( $2$ ).

$x^2 + 6x + 2 = 0$

Step 2: Halve the $x$ -coefficient ( $6$ ) and complete the square.

$(x + 3)^2 - 3^2 + 2 = 0$

Step 3: Simplify the constants.

$(x + 3)^2 - 9 + 2 = 0$
$(x + 3)^2 - 7 = 0$

Step 4: Isolate the bracket and square root both sides.

$(x + 3)^2 = 7$
$x + 3 = \pm\sqrt{7}$

Step 5: Solve for $x$ .

$x = -3 \pm \sqrt{7}$

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often square a negative half-coefficient incorrectly (e.g., thinking $(-3)^2$ is $-9$ ). The squared number is always positive, but you must still SUBTRACT this positive value outside the bracket.
2
In Edexcel exams, if a question asks for the 'exact form' or 'surd form', providing a decimal from your calculator will score zero accuracy marks.
3
Do not expand the brackets back out after completing the square to solve an equation; this completely reverses your work and defeats the purpose of the method.
4
You will typically earn a Method Mark (M1) just for showing the correct halved coefficient inside the bracket, e.g., $(x + 3)^2$ , even if you mess up the constant term.

Key Terms(8)

Completing the square: A method of rewriting a quadratic expression from $ax^2 + bx + c$ into the form $a(x + p)^2 + q$ .
Perfect square trinomial: A quadratic expression that can be factorised perfectly into a single squared bracket, such as $x^2 + 6x + 9 = (x + 3)^2$ .
Coefficient: The numerical value in front of a variable in an algebraic expression.
Quadratic rearrangement: The process of moving terms across the equals sign to set a quadratic equation to zero in the form $ax^2 + bx + c = 0$ .
Exact form: A requirement to provide answers using exact mathematical values like surds or fractions, rather than rounded decimals.
Surd form: A numerical result left with a root sign because the root is an irrational number.
Simplified surd form: A surd where the number under the root has no square factors other than 1 (e.g., $\sqrt{24}$ must be written as $2\sqrt{6}$ ).
Non-unitary coefficient: A coefficient of the $x^2$ term that is not equal to 1.

Previous NoteSolving Quadratic Equations: Factorising Next NoteSolving Quadratic Equations: The Quadratic Formula

Back to Quadratic Equations

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(8)

Completing the square: A method of rewriting a quadratic expression from $ax^2 + bx + c$ into the form $a(x + p)^2 + q$ .
Perfect square trinomial: A quadratic expression that can be factorised perfectly into a single squared bracket, such as $x^2 + 6x + 9 = (x + 3)^2$ .
Coefficient: The numerical value in front of a variable in an algebraic expression.
Quadratic rearrangement: The process of moving terms across the equals sign to set a quadratic equation to zero in the form $ax^2 + bx + c = 0$ .
Exact form: A requirement to provide answers using exact mathematical values like surds or fractions, rather than rounded decimals.
Surd form: A numerical result left with a root sign because the root is an irrational number.
Simplified surd form: A surd where the number under the root has no square factors other than 1 (e.g., $\sqrt{24}$ must be written as $2\sqrt{6}$ ).
Non-unitary coefficient: A coefficient of the $x^2$ term that is not equal to 1.

Solving Quadratic Equations: Completing the Square

The Power of Completing the Square

The "Half and Square" Rule

The core formula for completing the square is:

(x + \frac{b}{2})^2 - (\frac{b}{2})^2 + c

If the coefficient is negative, the bracket will be $(x - p)^2$ . Note that the square of a negative number is positive, but you must still subtract it outside the bracket.
If the coefficient is odd, leave the halved value as a fraction (e.g., $\frac{3}{2}$ ) rather than a decimal, as it makes squaring much easier (e.g., $(\frac{3}{2})^2 = \frac{9}{4}$ ).

Solving Quadratic Equations ( $a = 1$ )

When square-rooting, you must include the $\pm$ symbol to account for both the positive and negative root.
Edexcel Higher tier papers frequently specify that answers must be left in exact form or surd form (e.g., $x = a \pm b\sqrt{c}$ ) rather than decimals.
Often, you will need to leave your answer in simplified surd form, ensuring the number under the root has no hidden square factors.

Worked Example 1: Solving and Surd Simplification

Solve $2x^2 - 11x - 25 = x^2 - x$ by completing the square. Give your answer in simplified surd form.

Step 1: Rearrange the equation into the standard form $ax^2 + bx + c = 0$ .

$x^2 - 10x - 25 = 0$

Step 2: Halve the $x$ -coefficient ( $-10 \div 2 = -5$ ) to create the perfect square, then subtract its square.

$(x - 5)^2 - (-5)^2 - 25 = 0$

Step 3: Simplify the constant terms.

$(x - 5)^2 - 25 - 25 = 0$
$(x - 5)^2 - 50 = 0$

Step 4: Rearrange to isolate the squared bracket.

$(x - 5)^2 = 50$

Step 5: Square root both sides, remembering the $\pm$ symbol.

$x - 5 = \pm\sqrt{50}$

Step 6: Simplify the surd by extracting the largest square factor of $50$ ( $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$ ).

$x - 5 = \pm 5\sqrt{2}$

Step 7: Solve for $x$ .

$x = 5 \pm 5\sqrt{2}$

Handling Non-Unitary Coefficients

Worked Example 2: Solving with $a > 1$

Solve $2x^2 + 12x + 4 = 0$ by completing the square. Leave your answer in exact form.

Step 1: Divide the entire equation by the non-unitary coefficient ( $2$ ).

$x^2 + 6x + 2 = 0$

Step 2: Halve the $x$ -coefficient ( $6$ ) and complete the square.

$(x + 3)^2 - 3^2 + 2 = 0$

Step 3: Simplify the constants.

$(x + 3)^2 - 9 + 2 = 0$
$(x + 3)^2 - 7 = 0$

Step 4: Isolate the bracket and square root both sides.

$(x + 3)^2 = 7$
$x + 3 = \pm\sqrt{7}$

Step 5: Solve for $x$ .

$x = -3 \pm \sqrt{7}$

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often square a negative half-coefficient incorrectly (e.g., thinking $(-3)^2$ is $-9$ ). The squared number is always positive, but you must still SUBTRACT this positive value outside the bracket.
2
In Edexcel exams, if a question asks for the 'exact form' or 'surd form', providing a decimal from your calculator will score zero accuracy marks.
3
Do not expand the brackets back out after completing the square to solve an equation; this completely reverses your work and defeats the purpose of the method.
4
You will typically earn a Method Mark (M1) just for showing the correct halved coefficient inside the bracket, e.g., $(x + 3)^2$ , even if you mess up the constant term.

Key Terms(8)

Completing the square: A method of rewriting a quadratic expression from $ax^2 + bx + c$ into the form $a(x + p)^2 + q$ .
Perfect square trinomial: A quadratic expression that can be factorised perfectly into a single squared bracket, such as $x^2 + 6x + 9 = (x + 3)^2$ .
Coefficient: The numerical value in front of a variable in an algebraic expression.
Quadratic rearrangement: The process of moving terms across the equals sign to set a quadratic equation to zero in the form $ax^2 + bx + c = 0$ .
Exact form: A requirement to provide answers using exact mathematical values like surds or fractions, rather than rounded decimals.
Surd form: A numerical result left with a root sign because the root is an irrational number.
Simplified surd form: A surd where the number under the root has no square factors other than 1 (e.g., $\sqrt{24}$ must be written as $2\sqrt{6}$ ).
Non-unitary coefficient: A coefficient of the $x^2$ term that is not equal to 1.

Previous NoteSolving Quadratic Equations: Factorising Next NoteSolving Quadratic Equations: The Quadratic Formula

Back to Quadratic Equations

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(8)

Completing the square: A method of rewriting a quadratic expression from $ax^2 + bx + c$ into the form $a(x + p)^2 + q$ .
Perfect square trinomial: A quadratic expression that can be factorised perfectly into a single squared bracket, such as $x^2 + 6x + 9 = (x + 3)^2$ .
Coefficient: The numerical value in front of a variable in an algebraic expression.
Quadratic rearrangement: The process of moving terms across the equals sign to set a quadratic equation to zero in the form $ax^2 + bx + c = 0$ .
Exact form: A requirement to provide answers using exact mathematical values like surds or fractions, rather than rounded decimals.
Surd form: A numerical result left with a root sign because the root is an irrational number.
Simplified surd form: A surd where the number under the root has no square factors other than 1 (e.g., $\sqrt{24}$ must be written as $2\sqrt{6}$ ).
Non-unitary coefficient: A coefficient of the $x^2$ term that is not equal to 1.

Solving Quadratic Equations: Completing the Square

The Power of Completing the Square

The "Half and Square" Rule

Solving Quadratic Equations (a=1a = 1a=1)

Worked Example 1: Solving and Surd Simplification

Handling Non-Unitary Coefficients

Worked Example 2: Solving with a>1a > 1a>1

Sign up to continue reading

Exam Tips

Key Terms(8)

Key Terms(8)

Solving Quadratic Equations: Completing the Square

The Power of Completing the Square

The "Half and Square" Rule

Solving Quadratic Equations (a=1a = 1a=1)

Worked Example 1: Solving and Surd Simplification

Handling Non-Unitary Coefficients

Worked Example 2: Solving with a>1a > 1a>1

Sign up to continue reading

Exam Tips

Key Terms(8)

Key Terms(8)

Solving Quadratic Equations ( $a = 1$ )

Worked Example 2: Solving with $a > 1$

Solving Quadratic Equations ( $a = 1$ )

Worked Example 2: Solving with $a > 1$