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Simplification for SQA National 5 Maths

This page covers the following topics:

1. Collecting like terms
2. Expanding brackets
3. Completing the square

Algebraic expressions can be simplified by collecting like terms. Terms which are โ€œlikeโ€ are collected and the appropriate operations are applied to them to combine them. โ€œLikeโ€ terms are considered to be ones of the same variable with the same exponent.

Collecting like terms

Expanding brackets can be done when there is a term or a bracket in front of another bracket. When there is only a term in front of the bracket, every term in the bracket is multiplied by the one in front of it and the sum of the terms is taken to expand the bracket. When two brackets are being multiplied together, every term in the first bracket must be multiplied by every term in the second bracket and their sum must be taken to expand the brackets.

Expanding brackets

When a quadratic expression is not a perfect square, ie. it cannot be separated exactly into factors, it can be solved by completing the square. The square of an expression of the form axยฒ + bx + c can be completed by putting it in the form a((x + b/2a)ยฒ โˆ’ (b/2a)ยฒ + c/a). Setting the squared term to be equal to zero and solving for x gives the turning point of the quadratic equation.

Completing the square

1

Complete the square of the following expression: xยฒ + 6x + 11.

xยฒ + 6x + 11 = (x + 6/2)ยฒ โˆ’ (6/2)ยฒ + 11/1
= (x + 3)ยฒ โˆ’ 3ยฒ + 11
= (x + 3)ยฒ โˆ’ 9 + 11
= (x + 3)ยฒ + 2

(x + 3)ยฒ + 2

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2

Expand (2x + 1)(3x + 7).

(2x + 1)(3x + 7) =
= 2x ร— 3x + 2x ร— 7 + 1 ร— 3x + 1 ร— 7 =
= 6xยฒ + 14x + 3x + 7 =
= 6xยฒ + 17x + 7

6xยฒ + 17x + 7

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3

A rectangle has sides 3x and 4x + 1. Find a simplified expression for the area of the rectangle.

Area = 3x(4x + 1)
= 3x ร— 4x + 3x ร— 1
= 12xยฒ + 3x

12xยฒ + 3x

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