Expanding Quadratic Expressions
Expanding removes brackets by multiplying every term in one bracket by every term in the other, as in \((2x+3)(x-4)=2x^2-5x-12\). In NSW Year 10 Advanced maths this includes the special products \((a+b)^2\) and \((a-b)(a+b)\), surds, fractions and products of three or more terms (NESA outcome MA5‑ALG‑P‑02).
You will expand binomial products and perfect squares, simplify sums and differences of products, and expand expressions with surd or fraction coefficients, cubes and the square of a trinomial. You will find unknown coefficients, use a known square to find a value without solving, and write the revenue of a cafe as a quadratic. Harder questions prove facts about consecutive even numbers and sums of squares. It is part of Year 10 Advanced Quadratics.
Every question with a fully worked solution.
- Expanding Quadratic Expressions - Video - Expanding (multiplying out) double brackets Watch
Theory
To expand a product of brackets, multiply every term in one bracket by every term in the other, then collect like terms. This Year 10 Advanced maths page covers binomial products, the special products \((a + b)^2\) and \((a + b)(a - b)\), longer products and cubes, and using expansions to match coefficients and prove results.
A binomial has two terms, such as \(3x - 4\); a trinomial has three. A binomial product is two binomials multiplied, such as \((3x - 4)(x + 2)\).
Expanding removes the brackets using the distributive law: \(a(b + c) = ab + ac\). For two binomials, each of the two terms in the first multiplies each of the two in the second, giving four products. Then collect like terms.
Two patterns, the special products, come up so often that they are worth knowing by heart: the perfect square \((a + b)^2 = a^2 + 2ab + b^2\) and the difference of two squares \((a + b)(a - b) = a^2 - b^2\).
An identity is true for every value of the pronumeral, such as \((x + 1)^2 = x^2 + 2x + 1\). If two expansions are identical, their matching coefficients are equal.
How to expand and simplify
- Spot a special product first: a square \((a \pm b)^2\) or a sum times a difference \((a + b)(a - b)\).
- Multiply every term in one bracket by every term in the other. Watch the signs: a negative times a negative is positive.
- Collect like terms and write the answer in descending powers.
- Check by substituting a small number such as \(x = 1\) into both the question and the answer.
Harder expansions
- A product after a minus sign: expand it inside a bracket first, then subtract every term (Example 2).
- Three brackets or a cube: expand two brackets, then multiply by the third. Look for a difference of two squares to save work: \((x - 2)(x + 2)(x^2 + 4)\) \(= (x^2 - 4)(x^2 + 4) = x^4 - 16\).
- Area problems: write each area as a product, expand, then add or subtract.
- Matching coefficients: expand, then set the coefficients of \(x^2\), \(x\) and the constant equal (Example 4).
- Values without solving: expand a known square and substitute, e.g. \((p + q)^2 = p^2 + q^2 + 2pq\) gives \(p^2 + q^2\) from \(p + q\) and \(pq\).
- Proofs: write the numbers algebraically (consecutive integers \(n\), \(n + 1\); an odd number \(2n + 1\)), expand and simplify. For example \((n + 1)^2 - n^2 = 2n + 1\), so the difference of consecutive squares is always odd. A sum of squares is never negative, which proves inequalities.
(i) \((5x - 2)(3x + 4)\)
(ii) \((2a + 5b)^2\)
(iii) \((7 - 3y)(7 + 3y)\)
(i) Multiply each term in the first bracket by each term in the second, then collect the \(x\) terms.
| \((5x - 2)(3x + 4)\) | \(=\) | \(15x^2 + 20x - 6x - 8\) |
| \(\) | \(=\) | \(\boldsymbol{15x^2 + 14x - 8}\) |
(ii) Perfect square: first squared, twice the product, last squared.
| \((2a + 5b)^2\) | \(=\) | \((2a)^2 + 2(2a)(5b) + (5b)^2\) |
| \(\) | \(=\) | \(\boldsymbol{4a^2 + 20ab + 25b^2}\) |
(iii) Difference of two squares: the middle terms cancel.
| \((7 - 3y)(7 + 3y)\) | \(=\) | \(7^2 - (3y)^2\) |
| \(\) | \(=\) | \(\boldsymbol{49 - 9y^2}\) |
Expand each part separately. Keep the second product in a bracket so the minus sign reaches every term.
| \(3(x + 2)^2\) | \(=\) | \(3(x^2 + 4x + 4) = 3x^2 + 12x + 12\) |
| \((x - 4)(x + 5)\) | \(=\) | \(x^2 + x - 20\) |
| \(\) | \(\) | \(3x^2 + 12x + 12 - (x^2 + x - 20)\) |
| \(\) | \(=\) | \(3x^2 + 12x + 12 - x^2 - x + 20\) |
| \(\) | \(=\) | \(\boldsymbol{2x^2 + 11x + 32}\) |
Check with \(x = 1\): \(3(9) - (-3)(6) = 27 + 18 = 45\), and \(2 + 11 + 32 = 45\).
(i) \((2x - 1)(x^2 + 4x - 3)\)
(ii) \((x - 2)^3\)
(i) Multiply the whole trinomial by \(2x\), then by \(-1\), and line up like terms.
| \(2x(x^2 + 4x - 3)\) | \(=\) | \(2x^3 + 8x^2 - 6x\) |
| \(-1(x^2 + 4x - 3)\) | \(=\) | \(-x^2 - 4x + 3\) |
| \(\text{sum}\) | \(=\) | \(\boldsymbol{2x^3 + 7x^2 - 10x + 3}\) |
(ii) Write the cube as a square times a bracket.
| \((x - 2)^3\) | \(=\) | \((x - 2)^2(x - 2)\) |
| \(\) | \(=\) | \((x^2 - 4x + 4)(x - 2)\) |
| \(\) | \(=\) | \(x^3 - 2x^2 - 4x^2 + 8x + 4x - 8\) |
| \(\) | \(=\) | \(\boldsymbol{x^3 - 6x^2 + 12x - 8}\) |
Count the terms before collecting: \(2 \times 3 = 6\) products in (i).
Expand the left side, then make each coefficient match the right side.
| \((px - 2)(3x + q)\) | \(=\) | \(3px^2 + pqx - 6x - 2q\) |
| \(\) | \(=\) | \(3px^2 + (pq - 6)x - 2q\) |
| \(x^2{:}\quad 3p\) | \(=\) | \(12, \text{ so } p = \boldsymbol{4}\) |
| \(\text{constant}{:}\quad -2q\) | \(=\) | \(10, \text{ so } q = \boldsymbol{-5}\) |
| \(x{:}\quad r\) | \(=\) | \(pq - 6 = 4(-5) - 6 = \boldsymbol{-26}\) |
Check: \((4x - 2)(3x - 5) = 12x^2 - 20x - 6x + 10 = 12x^2 - 26x + 10\).
Common pitfalls
Frequently asked questions
How do you expand two brackets?
Multiply each term in the first bracket by each term in the second bracket, which gives four products for two binomials. Then collect the like terms, usually the two x terms.
What is the formula for (a + b) squared?
\((a + b)^2 = a^2 + 2ab + b^2\). Square the first term, double the product of the two terms, and square the last term.
What is the difference of two squares?
It is the pattern \((a + b)(a - b) = a^2 - b^2\). The middle terms cancel because one is \(+ab\) and the other is \(-ab\).
How do you expand (x + 2) cubed?
Write it as \((x + 2)^2(x + 2)\). Expand the square to get \(x^2 + 4x + 4\), then multiply by \(x + 2\) and collect like terms: \(x^3 + 6x^2 + 12x + 8\).
How do you check an expansion?
Substitute a simple value such as x = 1 or x = 2 into the original expression and into your answer. If the two values differ, there is an error.