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Worked solution

Complete the square

Express the following quadratic functions in the form a(x+p)2+qa(x+p)^2+q, where aa, pp and qq are constants to be found.

(a) 2x2−4x+52x^2-4x+5

(b) 24−9x−x224-9x-x^2

CJC · Discussion · Question 2

Answer

(a) 2(x−1)2+32(x-1)^2+3. (b) −(x+9/2)2+177/4-(x+9/2)^2+177/4.

(a) Factor out 2 before completing the square

2x2−4x+5=2(x2−2x)+5=2[(x−1)2−1]+5=2(x−1)2+32x^2-4x+5=2(x^2-2x)+5=2[(x-1)^2-1]+5=2(x-1)^2+3. Thus a=2a=2, p=−1p=-1, q=3q=3.

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(b) Factor out the negative sign

24−9x−x2=−(x2+9x)+24=−[(x+9/2)2−(9/2)2]+24=−(x+9/2)2+177/424-9x-x^2=-(x^2+9x)+24=-[(x+9/2)^2-(9/2)^2]+24=-(x+9/2)^2+177/4. Thus a=−1a=-1, p=9/2p=9/2, q=177/4q=177/4.

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Common mistakes

Keep the outside coefficient when compensating for the added square.

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