Question

5.1 cool down
question 2:
solve the equation ( x^2 - 16x = -60 ) by completing the square. show your reasoning.

fill in the blank 2 points
question 2:
solve the equation ( x^2 - 16x = -60 ) by completing the square.
write the smallest value first. if only one solution, write n/a in the second blank.
( x = ) type your answer..
or ( x = ) type your answer..

essay 10 points
question 2:
show your reasoning. (show your work for credit, must be completing the square method.)

Explanation:

Step1: Start with the given equation

The equation is \( x^2 - 16x = -60 \). To complete the square, we take the coefficient of \( x \), which is \(-16\), divide it by \( 2 \) to get \(-8\), and then square it: \( (-8)^2 = 64 \).

Step2: Add 64 to both sides

Add \( 64 \) to both sides of the equation:
\[
x^2 - 16x + 64 = -60 + 64
\]
The left - hand side is now a perfect square trinomial, which factors as \( (x - 8)^2 \), and the right - hand side simplifies to \( 4 \):
\[
(x - 8)^2 = 4
\]

Step3: Take the square root of both sides

Take the square root of both sides. Remember that when we take the square root of a number, we get both a positive and a negative root:
\[
x - 8=\pm\sqrt{4}=\pm2
\]

Step4: Solve for x

Case 1: When \( x - 8 = 2 \)
Add \( 8 \) to both sides: \( x=2 + 8=10 \)

Case 2: When \( x - 8=-2 \)
Add \( 8 \) to both sides: \( x=-2 + 8 = 6 \)

Answer:

The solutions are \( x = 6 \) or \( x = 10 \) (the smaller value first, so \( x = 6 \) and \( x = 10 \))