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Paper 1 · November 2010 · Integration and Reduction Formulae

Express $4x^2 - 4x + 5$ in the form $(ax + b)^2 + c$ , where $a$ , $b$ and $c$ are integers.

Model answer

(2x-1)^2+4

Also accepted: (2x - 1)^2 + 4, (2x-1)^2 + 4

Explanation

Matching the

x^2

term,

a^2=4

so take

a=2

. Then

(2x+b)^2=4x^2+4bx+b^2

, and matching the

x

term gives

4b=-4

, so

b=-1

. That leaves

b^2=1

, so

c=5-1=4

, giving

4x^2-4x+5=(2x-1)^2+4

.

Derived from ZIMSEC Further Mathematics 9187/1, November 2010, Q13

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