Perfect Squares and Difference of Perfect Squares
Theory
To expand a product of brackets, multiply every term in one bracket by every term in the other, then collect like terms. Three products come up so often that they are worth knowing by heart: the perfect squares \((a + b)^2\) and \((a - b)^2\), and the difference of perfect squares \((a + b)(a - b)\). This Year 9 maths page shows how to expand and simplify them quickly and accurately, and how to use them to calculate and solve problems.
A binomial is an expression with two terms, such as \(x + 5\) or \(3a - 2b\). A binomial product is two binomials multiplied: \((x + 2)(x - 7)\).
Expanding removes the brackets. The result is usually a quadratic expression: its highest power is \(2\), as in \(2x^2 + 5x - 3\).
A perfect square is a binomial times itself, \((a + b)^2 = (a + b)(a + b)\). A difference of two squares comes from \((a + b)(a - b)\): the middle terms cancel.
Two expressions are identical when they are equal for every value of the pronumeral. Then the coefficients of each power must match, which lets you find unknown constants.
Here \(a\) and \(b\) can be any terms: \(3x\), \(5y\), \(\dfrac{x}{2}\), or even a whole bracket.
How to expand and simplify
- Spot a special product first: a square, or the same two terms with opposite signs.
- Otherwise multiply every term in the first bracket by every term in the second (four products).
- Several products? Expand each one inside a bracket, then multiply by the number in front. A minus in front changes every sign in that bracket.
- Collect like terms and write the answer in descending powers.
- Check by substituting a small value such as \(x = 1\) into both the question and the answer.
Using expansions
- Mental arithmetic: write a number near a round number, \(97 = 100 - 3\), and expand.
- Matching coefficients: expand one side, then set the \(x^2\), \(x\) and constant coefficients equal to find unknowns.
- Areas and proofs: write each length as an expression, expand, and simplify. If the pronumeral cancels, the result is the same for every value.
(i) \((4x + 5)^2\)
(ii) \((3a - 2b)^2\)
(iii) \((6m - 1)(6m + 1)\)
(iv) \(\left(\dfrac{x}{3} + 2\right)^2\)
(i) and (ii) are perfect squares: square the first term, double the product, square the last term.
| \((4x + 5)^2\) | \(=\) | \((4x)^2 + 2(4x)(5) + 5^2\) |
| \(\) | \(=\) | \(\boldsymbol{16x^2 + 40x + 25}\) |
| \((3a - 2b)^2\) | \(=\) | \((3a)^2 - 2(3a)(2b) + (2b)^2\) |
| \(\) | \(=\) | \(\boldsymbol{9a^2 - 12ab + 4b^2}\) |
(iii) Same terms, opposite signs: a difference of two squares. The middle terms cancel.
| \((6m - 1)(6m + 1)\) | \(=\) | \((6m)^2 - 1^2 = \boldsymbol{36m^2 - 1}\) |
(iv) The same pattern works with a fraction.
| \(\left(\dfrac{x}{3} + 2\right)^2\) | \(=\) | \(\dfrac{x^2}{9} + 2 \times \dfrac{x}{3} \times 2 + 4\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{x^2}{9} + \dfrac{4x}{3} + 4}\) |
(i) \(2(x - 3)^2 - (x + 4)(x - 4)\)
(ii) \(-(3x + 2)(x - 4)\)
(i) Expand each product inside brackets first, then multiply by what is in front. The minus sign changes every sign in the second product.
| \(2(x - 3)^2\) | \(=\) | \(2(x^2 - 6x + 9)\) |
| \(\) | \(=\) | \(2x^2 - 12x + 18\) |
| \((x + 4)(x - 4)\) | \(=\) | \(x^2 - 16\) |
| \(\text{so the whole}\) | \(=\) | \(2x^2 - 12x + 18 - (x^2 - 16)\) |
| \(\) | \(=\) | \(2x^2 - 12x + 18 - x^2 + 16\) |
| \(\) | \(=\) | \(\boldsymbol{x^2 - 12x + 34}\) |
(ii) Expand the product, then change the sign of every term.
| \((3x + 2)(x - 4)\) | \(=\) | \(3x^2 - 12x + 2x - 8\) |
| \(\) | \(=\) | \(3x^2 - 10x - 8\) |
| \(-(3x + 2)(x - 4)\) | \(=\) | \(\boldsymbol{-3x^2 + 10x + 8}\) |
Check with \(x = 1\): \(-(5)(-3) = 15\) and \(-3 + 10 + 8 = 15\ \checkmark\)
(i) Without a calculator, evaluate \(52^2\) and \(49 \times 51\).
(ii) Find \(a\) and \(b\) if \((x - a)^2 = x^2 - 18x + b\).
(iii) Find \(k\) if \((3x + k)(x + 2) = 3x^2 + 11x + 10\) for all \(x\).
(i) Split each number around a round number.
| \(52^2\) | \(=\) | \((50 + 2)^2\) |
| \(=\) | \(2500 + 200 + 4 = \boldsymbol{2704}\) | |
| \(49 \times 51\) | \(=\) | \((50 - 1)(50 + 1)\) |
| \(=\) | \(2500 - 1 = \boldsymbol{2499}\) |
(ii) Expand the left side and match the coefficients.
| \((x - a)^2\) | \(=\) | \(x^2 - 2ax + a^2\) |
| \(-2a\) | \(=\) | \(-18 \ \Rightarrow\ \boldsymbol{a = 9}\) |
| \(b\) | \(=\) | \(a^2 = \boldsymbol{81}\) |
(iii) Expand and compare the \(x\) terms.
| \((3x + k)(x + 2)\) | \(=\) | \(3x^2 + (6 + k)x + 2k\) |
| \(6 + k\) | \(=\) | \(11 \ \Rightarrow\ \boldsymbol{k = 5}\) |
Check the constant term: \(2k = 10\ \checkmark\)
(i) The path adds \(1\) m on both ends of each side, so the outer rectangle is \((2x + 2)\) m by \((x + 5)\) m.
| \(\text{outer}\) | \(=\) | \((2x + 2)(x + 5)\) |
| \(\) | \(=\) | \(2x^2 + 12x + 10\) |
| \(\text{garden}\) | \(=\) | \(2x(x + 3) = 2x^2 + 6x\) |
| \(\text{path}\) | \(=\) | \(2x^2 + 12x + 10 - (2x^2 + 6x)\) |
| \(\) | \(=\) | \(\boldsymbol{6x + 10}~\text{m}^2\) |
| \(6(4) + 10\) | \(=\) | \(\boldsymbol{34}~\text{m}^2\) |
(ii) Check with lengths: outer \(10 \times 9 = 90\), garden \(8 \times 7 = 56\), path \(90 - 56 = 34\ \checkmark\)
Common pitfalls
Frequently asked questions
How do you expand a binomial product?
Multiply each term in the first bracket by each term in the second bracket, giving four products, then collect like terms.
What is the rule for expanding a perfect square?
Square the first term, add twice the product of the two terms, then add the square of the last term: \((a + b)^2 = a^2 + 2ab + b^2\).
What is the difference of two squares?
The product \((a + b)(a - b)\) equals \(a^2 - b^2\), because the two middle terms cancel.
How do you expand when there is a minus sign in front of the brackets?
Expand the product first and keep the result in brackets. Then the minus sign changes the sign of every term inside.
How can expanding help with mental arithmetic?
Write a number as a round number plus or minus a small one. For example \(99^2\) \(= (100 - 1)^2\) \(= 10\,000 - 200 + 1\) \(= 9801\).
How do you find an unknown constant in an expansion?
Expand the side with the unknown, then match the coefficients of each power of \(x\) on both sides and solve.