Question

Consider the quadratic equation x2 − 6x = 16.

Which is equivalent to the given quadratic equation?
A) (x + 4)2 = 16
B) (x − 4)2 = 16
C) (x − 3)2 = 25
D) (x + 3)2 = 16

AND

Suppose you are completing the square to rewrite 4x2 + 40x = 80 in the form a(x − p)2 = q. What is the first step to take?
A) Square half of 40 and add the result to both sides of the equation.
B) Subtract 80 from both sides of the equation.
C) Factor out 4 from 4x2 + 40x.
D) Set the equation equal to zero.

Answer 1

1. C) (x − 3)2 = 25
2. C) Factor out 4 from 4x2 + 40x.

Explanation:

1. Adding the square of half the x-coefficient to both sides of the equation will "complete the square." That square is 9, so the result on the right is 16+9 = 25. Only selection C matches.

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2. To complete the square, you want to be able to put the quadratic into the form a(x -h)^2 = -k. For the purpose, it is most convenient to first factor "a" from the given quadratic. Then you can determine "-h" to be half the x-coefficient inside the parentheses.

Here, that looks like ...

4(x² +10x) = 80 . . . . . . . . . . step 1: factor out 4

4(x² +10x +25) = 180 . . . . . add 25 inside parentheses and the same number (4·25) on the right side of the equation

4(x +5)² = 180 . . . . . . . . . . . written as a square