Expanding Expressions
Every question with a fully worked solution.
- Expanding Expressions - Video - Expanding (multiplying out) double brackets Watch
Theory
This Year 10 maths page shows how to expand binomial products such as \((x + 4)(x - 7)\), how to spot the special products \((a + b)^2\), \((a - b)^2\) and \((a + b)(a - b)\), and how to expand and simplify longer expressions, including a minus sign in front of a product, fractions, and a binomial times a trinomial. These skills lead straight into factorising and solving quadratic equations.
To expand means to remove the brackets by multiplying. A binomial is an expression with two terms, such as \(x + 4\). A binomial product is two binomials multiplied together, such as \((x + 4)(x - 7)\).
Every term in the first bracket multiplies every term in the second, so a binomial product gives four terms before simplifying. The two middle terms are often like terms, so they combine. The sign in front of a term belongs to it.
A perfect square is a binomial multiplied by itself, such as \((x + 3)^2 = (x + 3)(x + 3)\).
A difference of two squares comes from two brackets that differ only in their sign, such as \((x + 3)(x - 3) = x^2 - 9\). The middle terms cancel.
A number in front, as in \(3(x + 1)(x - 2)\), multiplies the whole product. Expand the brackets first, then multiply every term by the number.
The distributive law for a binomial product:
The special products:
How to expand a binomial product
- Multiply the first term of the first bracket by each term of the second bracket.
- Multiply the second term of the first bracket by each term of the second bracket, keeping its sign.
- Collect the like terms, usually the two middle terms.
- Order the answer: the squared term, then the \(x\) term, then the number.
How to expand and simplify a longer expression
- Look for patterns first: a perfect square or a difference of two squares can be written down quickly.
- Expand each product on its own. If there is a number in front, expand the brackets first, then multiply by the number.
- Keep brackets around a product that is being subtracted, then change every sign inside when the brackets come off.
- Collect like terms, then check by substituting a small number such as \(x = 2\) into the question and the answer.
Three or more factors, and groups
- Three brackets: expand two of them first, then multiply by the third. Pair up a difference of two squares if you can: \((x - 2)(x + 2)(x^2 + 4)\) \(= (x^2 - 4)(x^2 + 4) = x^4 - 16\).
- A group as one term: in \((a + b + 1)(a + b - 1)\) treat \(a + b\) as one term. It is a difference of two squares: \((a + b)^2 - 1 = a^2 + 2ab + b^2 - 1\).
- Square numbers in your head: write a number near \(100\) as \(100 + n\) or \(100 - n\) and expand, as in the Formulas tab.
How to find an unknown coefficient
- Expand the side with the unknown.
- Match the numbers (or the \(x\) terms) on both sides and solve for the unknown. Check with the other terms.
(i) \((y + 5)(y - 8)\)
(ii) \((4x - 3)(2x + 7)\)
Multiply each term in the first bracket by each term in the second, then collect the like terms.
(i)
| \((y + 5)(y - 8)\) | \(=\) | \(y^2 - 8y + 5y - 40\) |
| \(\) | \(=\) | \(\boldsymbol{y^2 - 3y - 40}\) |
(ii) Multiply the first terms, the outer terms, the inner terms and the last terms.
\((4x - 3)(2x + 7)\)
| \(\) | \(=\) | \(8x^2 + 28x - 6x - 21\) |
| \(\) | \(=\) | \(\boldsymbol{8x^2 + 22x - 21}\) |
(i) \((2a - 5)^2\)
(ii) \((6m + 1)(6m - 1)\)
(i) A perfect square: \((A - B)^2 = A^2 - 2AB + B^2\) with \(A = 2a\) and \(B = 5\).
| \((2a - 5)^2\) | \(=\) | \((2a)^2 - 2 \times 2a \times 5 + 5^2\) |
| \(\) | \(=\) | \(\boldsymbol{4a^2 - 20a + 25}\) |
(ii) The brackets are the same except for the sign, so this is a difference of two squares. The middle terms cancel.
\((6m + 1)(6m - 1)\)
| \(\) | \(=\) | \(36m^2 - 6m + 6m - 1\) |
| \(\) | \(=\) | \(\boldsymbol{36m^2 - 1}\) |
Expand the binomial product first, then multiply by \(2\). Keep the square in brackets: the minus sign applies to all of it.
| \(2(x - 4)(x + 1)\) | \(=\) | \(2(x^2 + x - 4x - 4)\) |
| \(\) | \(=\) | \(2x^2 - 6x - 8\) |
| \((x - 3)^2\) | \(=\) | \(x^2 - 6x + 9\) |
Subtract the whole second result:
| \(\) | \(\) | \((2x^2 - 6x - 8) - (x^2 - 6x + 9)\) |
| \(\) | \(=\) | \(2x^2 - 6x - 8 - x^2 + 6x - 9\) |
| \(\) | \(=\) | \(\boldsymbol{x^2 - 17}\) |
Check with \(x = 2\): the question gives \(2 \times (-2) \times 3 - 1 = -13\) and the answer gives \(4 - 17 = -13\).
Expand the left side, then match it with the right side term by term.
\((3x - 2)(x + k)\)
| \(\) | \(=\) | \(3x^2 + 3kx - 2x - 2k\) |
The numbers on their own must match:
| \(-2k\) | \(=\) | \(-10\) |
| \(k\) | \(=\) | \(\boldsymbol{5}\) |
Check the \(x\) terms: \(3kx - 2x = 15x - 2x = 13x\). ✓
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, keeping the signs. Then collect the like terms, which are usually the two middle terms.
What is a perfect square?
A perfect square is a binomial multiplied by itself. \((a + b)^2 = a^2 + 2ab + b^2\), so the middle term is double the product of the two terms.
What is the difference of two squares?
It is the product of two brackets that are the same except for the sign, such as \((a + b)(a - b)\). The middle terms cancel, leaving \(a^2 - b^2\).
How can I check an expansion?
Substitute a small number, such as \(x = 2\), into the question and into your answer. If the two values are different, there is a mistake.
Why are binomial products important?
They are the reverse of factorising quadratics. Expanding and factorising are used to solve quadratic equations and to find the features of parabolas.