 # Simplification for SQA Higher Maths

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 ax² + bx + c can be completed by putting it in the form a((x + b/2a)² − (b/2a)² + c/a). Setting the squared term to be equal to zero and solving for x gives the turning point of the quadratic equation. # 1

Complete the square of the following expression: x² + 6x + 11.

x² + 6x + 11 = (x + 6/2)² − (6/2)² + 11/1
= (x + 3)² − 3² + 11
= (x + 3)² − 9 + 11
= (x + 3)² + 2

(x + 3)² + 2 # 2

Express the following in the completed square form: y = 8x² + 32x + 160.

y = 8x² + 32x + 160
y = 8(x² + 4x + 20)
y = 8((x + 4/2)² − (4/2)² + 20)
y = 8((x + 2)² − 2² + 20)
y = 8((x + 2)² − 4 + 20)
y = 8((x + 2)² + 16)
y = 8(x + 2)² + 128

y = 8(x + 2)² + 128 # 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 × 3x + x × 1 + 2 × 3x + 2 × 1
= 3x² + x + 6x + 2
= 3x² + 7x + 2
= 3(x² + 7x/3 + 2/3)
= 3((x + 7/6)² − (7/6)² + 2/3)
= 3((x + 7/6)² − 49/36 + 2/3)
= 3((x + 7/6)² − 25/36)
= 3(x + 7/6)² − 25/12

3(x + 7/6)² − 25/12 End of page