IGCSE Maths / Advanced algebra
Expanding brackets with surds
Edexcel International GCSE Maths (4MA1), chapter 23: Advanced algebra, specification point 1.4b. Free lesson.
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Expand brackets containing surds by making the same four multiplications as for any double bracket. Use √a × √a = a, keep each sign with its root, then collect like surds and plain numbers.
How to expand brackets with surds
A surd is a root that is not a whole number, such as √5. Expanding means multiplying everything out until there are no brackets left. Use the same four-box grid as for any pair of double brackets. The special rule is that a root multiplied by itself becomes the number inside the root. For example, √5 × √5 = 5.
Expand and collect the terms
In (3 + √5)(2 − √5), the minus sign belongs to −√5, so it must stay with that term in the grid. The two middle products are like surds, so they can be collected. The root multiplied by itself gives a plain number. The root terms combine as −3√5 + 2√5 = −√5. The plain numbers combine as 6 − 5 = 1.
Keep the sign when roots meet
When one root is positive and the other is negative, their product is negative. The roots multiply to give the number inside, while the single minus sign remains. This is a common place to lose a mark. √5 × −√5 = −5, not +5. Two minus signs would make a positive product, but this multiplication has only one.
Square a bracket by multiplying it by itself
Squaring a bracket means multiplying the entire bracket by itself, so there are still four products. The two middle products may be the same, and then they add together. Do not square only the terms and leave out the middle products. For (2 + √3)², the middle products are both 2√3, so together they make 4√3.
Recognise when the middle terms cancel
If two brackets have the same terms but opposite signs between them, the middle products have equal sizes and opposite signs. They cancel, leaving only the plain numbers. In the worked example, this gives a whole number with no surd left. For (4 + √7)(4 − √7), −4√7 and +4√7 cancel to 0. The remaining numbers are 16 − 7 = 9.