Question

which expression is equivalent to $x^2 - 16x + 49$?
a $(x - 7)^2$
b $(x - 8)^2$
c $(x - 8)^2 - 15$
d $(x - 8)^2 + 41$

Explanation:

Step1: Recall the perfect square formula

The perfect square formula is \((a - b)^2 = a^2 - 2ab + b^2\). For the quadratic expression \(x^2 - 16x + 49\), we want to express it in the form of a perfect square plus or minus a constant.

Step2: Complete the square for the quadratic expression

For the quadratic term \(x^2 - 16x\), to complete the square, we take half of the coefficient of \(x\), which is \(\frac{-16}{2}=-8\), and then square it: \((-8)^2 = 64\). So we can rewrite \(x^2 - 16x\) as \((x - 8)^2 - 64\) (because \((x - 8)^2=x^2 - 16x + 64\), so to get \(x^2 - 16x\), we subtract 64).

Step3: Substitute back into the original expression

Now substitute \(x^2 - 16x=(x - 8)^2 - 64\) into the original expression \(x^2 - 16x + 49\):

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Answer:

C. \((x - 8)^2 - 15\)