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Solution 2.3:1b

From Förberedande kurs i matematik 1

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When we complete the square, it is only the first two terms, x2+2x, that are involved. The general formula for completing the square states that x2+ax equals

x+2a2−2a2. 

Note how the coefficient a in front of the x turns up halved in two places.

If we use this formula, we obtain

x2+2x=x+222−222=(x+1)2−1 

and if we subtract the last "1", we obtain

x2+2x−1=(x+1)2−1−1=(x+1)2−2.

To be completely certain that we have used the correct formula, we can expand the quadratic on the right-hand side,

(x+1)2−2=x2+2x+1−2=x2+2x−1

and see that the relation really holds.

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