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Simplification for SQA Higher Maths

This page covers the following topics:

1. Completing the square

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

Express the following in the completed square form: y = 8xยฒ + 32x + 160.

y = 8xยฒ + 32x + 160
y = 8(xยฒ + 4x + 20)
y = 8((x + 4/2)ยฒ โˆ’ (4/2)ยฒ + 20)
y = 8((x + 2)ยฒ โˆ’ 2ยฒ + 20)
y = 8((x + 2)ยฒ โˆ’ 4 + 20)
y = 8((x + 2)ยฒ + 16)
y = 8(x + 2)ยฒ + 128

y = 8(x + 2)ยฒ + 128

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3

Laura finds that her rectangular poster has dimensions x + 2 and 3x + 1. Find an expression for the area of the poster and complete the square.

Area = (x + 2)(3x + 1)
= x ร— 3x + x ร— 1 + 2 ร— 3x + 2 ร— 1
= 3xยฒ + x + 6x + 2
= 3xยฒ + 7x + 2
= 3(xยฒ + 7x/3 + 2/3)
= 3((x + 7/6)ยฒ โˆ’ (7/6)ยฒ + 2/3)
= 3((x + 7/6)ยฒ โˆ’ 49/36 + 2/3)
= 3((x + 7/6)ยฒ โˆ’ 25/36)
= 3(x + 7/6)ยฒ โˆ’ 25/12

3(x + 7/6)ยฒ โˆ’ 25/12

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