Question

1. which of the following equations is equivalent to

$x^2 - 12x = 28$

a) $(x - 6)^2 = 36$
b) $(x - 6)^2 = 64$
c) $(x - 6)^2 = 28$
d) $(x - 6)^2 = 100$

Explanation:

Step1: Recall Completing the Square

To solve \(x^2 - 12x = 28\) by completing the square, take half of the coefficient of \(x\), square it, and add to both sides. The coefficient of \(x\) is \(-12\), half of it is \(-6\), and squaring gives \((-6)^2 = 36\).

Step2: Apply Completing the Square

Add \(36\) to both sides of \(x^2 - 12x = 28\):
\(x^2 - 12x + 36 = 28 + 36\)
The left side factors as \((x - 6)^2\), and the right side is \(64\) (since \(28 + 36 = 64\)). So the equation becomes \((x - 6)^2 = 64\), which matches option b.

Answer:

b) \((x - 6)^2 = 64\)