Question

determine the ( x )-intercepts of the quadratic function with equation ( y = -(x - 7)^2 + 4 ) separate your answers with a comma.

Explanation:

Step1: Set \( y = 0 \)

To find the \( x \)-intercepts, we set \( y = 0 \) in the equation \( y = -(x - 7)^2 + 4 \). So we have the equation:
\[
0 = -(x - 7)^2 + 4
\]

Step2: Rearrange the equation

First, we can rewrite the equation as:
\[
(x - 7)^2 = 4
\]
This is because we add \( (x - 7)^2 \) to both sides and subtract 0 (which doesn't change the equation) and then we get \( (x - 7)^2 = 4 \).

Step3: Take square roots

Taking the square root of both sides, we know that if \( a^2 = b \), then \( a=\pm\sqrt{b} \). So for \( (x - 7)^2 = 4 \), we have:
\[
x - 7=\pm\sqrt{4}
\]
Since \( \sqrt{4} = 2 \), this simplifies to:
\[
x - 7=\pm2
\]

Step4: Solve for \( x \)

We now solve for \( x \) in two cases.

Case 1: \( x - 7 = 2 \)
Adding 7 to both sides, we get:
\[
x=2 + 7=9
\]

Case 2: \( x - 7=- 2 \)
Adding 7 to both sides, we get:
\[
x=-2 + 7 = 5
\]

Answer:

5, 9