Completing the square Revision Notes for AQA GCSE Further Maths

Completing the square

What is completing the square?

Completing the square is a fundamental algebraic technique used to rewrite quadratic expressions in a special form. When working with a quadratic expression, it can be very useful to write it to include the term $(x + a)^2$ or $(x - a)^2$ , where $a$ is a constant. This approach has important applications in solving quadratic equations and working with quadratic graphs.

The main goal is to transform a quadratic expression like $x^2 + bx + c$ into the form $(x + p)^2 + q$ , where $p$ and $q$ are constants we need to find.

The basic method

The technique involves expanding brackets and then comparing coefficients on both sides of an equation. When we say "equate coefficients," we mean making equal the number of terms with the same power on each side of the identity.

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Key Algebraic Identity

Remember that when you expand $(x - p)^2$ , you get: $(x - p)^2 = x^2 - 2px + p^2$

This expansion is crucial for understanding how to work backwards from a quadratic expression to its completed square form.

Worked example 1: Finding p and q

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Worked Example: Finding p and q values

Let's work through finding the values of $p$ and $q$ when $x^2 - 6x + 2 = (x - p)^2 + q$ .

Step 1: Expand the right side Start by expanding $(x - p)^2$ : $(x - p)^2 + q = x^2 - 2px + p^2 + q$

Step 2: Compare with the original expression Now we have: $x^2 - 6x + 2 = x^2 - 2px + p^2 + q$

Step 3: Equate coefficients of x Looking at the $x$ terms: $-6 = -2p$ Solving this: $p = 3$

Step 4: Equate the constant terms Looking at the constants: $2 = p^2 + q$ Substituting $p = 3$ : $2 = 9 + q$ Therefore: $q = -7$

Final answer: $p = 3$ and $q = -7$

Worked example 2: Different coefficient forms

When working with expressions that have coefficients other than 1 for $x^2$ , the method remains the same but requires careful attention to all coefficients.

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Worked Example: Handling different coefficients

For the expression $2x^2 + bx + 5 = a(x - 3)^2 + c$ :

Step 1: Expand the right side $a(x - 3)^2 + c = a(x^2 - 6x + 9) + c = ax^2 - 6ax + 9a + c$

Step 2: Compare coefficients

Coefficients of $x^2$ : $2 = a$
Coefficients of $x$ : $b = -6a = -6(2) = -12$
Constant terms: $5 = 9a + c = 9(2) + c = 18 + c$ , so $c = -13$

Final answer: $a = 2$ , $b = -12$ , and $c = -13$

Worked example 3: Working with positive coefficients inside brackets

For expressions with positive signs inside brackets, the approach remains systematic but requires attention to the sign changes.

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Worked Example: Positive coefficients in brackets

For $3x^2 + 5x - 1 = a(x + b)^2 + c$ , notice the positive sign inside the brackets:

Step 1: Expand $a(x + b)^2 + c = a(x^2 + 2bx + b^2) + c = ax^2 + 2abx + ab^2 + c$

Step 2: Equate coefficients systematically

$x^2$ coefficients: $3 = a$
$x$ coefficients: $5 = 2ab$ , so $5 = 2(3)b = 6b$ , giving $b = \frac{5}{6}$
Constants: $-1 = ab^2 + c$

Step 3: Find c $-1 = 3 \times \left(\frac{5}{6}\right)^2 + c$ $-1 = 3 \times \frac{25}{36} + c$ $-1 = \frac{25}{12} + c$ $c = -1 - \frac{25}{12} = -\frac{37}{12}$

Final answer: $a = 3$ , $b = \frac{5}{6}$ , and $c = -\frac{37}{12}$

Key technique: Comparing coefficients

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Comparing Coefficients Method

Comparing coefficients is a powerful technique that can be applied to any polynomial. The method involves:

Expanding any brackets on one or both sides
Collecting like terms together
Matching coefficients of the same powers of $x$
Solving the resulting system of equations

This systematic approach ensures you don't miss any terms and helps avoid common algebraic errors.

Common exam approaches

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Critical Exam Tips

When tackling completing the square problems:

Always expand brackets fully before comparing
Be systematic about equating coefficients - start with the highest power and work down
Double-check your arithmetic, especially when working with fractions
Remember that the constant term includes all numbers without $x$
Pay attention to signs, particularly when dealing with $(x - p)$ versus $(x + p)$ forms

Connection to quadratic equations

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While we'll explore this further in later topics, completing the square is fundamental to solving quadratic equations and understanding quadratic graphs. The ability to rewrite expressions in the form $a(x + b)^2 + c$ is a key skill that appears throughout mathematics.

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Key Points to Remember:

Completing the square transforms quadratic expressions into the form $(x \pm p)^2 + q$
The key technique is expanding brackets and equating coefficients of like terms
"Equating coefficients" means making the coefficients of the same powers of $x$ equal on both sides
Always work systematically: $x^2$ terms first, then $x$ terms, then constants
The standard expansion $(x - p)^2 = x^2 - 2px + p^2$ is essential to remember

Completing the square (AQA GCSE Further Maths): Revision Notes