To find the inverse of the given equation [tex]\( 2(x-2)^2 = 8(7+y) \)[/tex], we need to isolate [tex]\( y \)[/tex] and express it in terms of [tex]\( x \)[/tex].
Let's start solving the given equation step-by-step:
1. Given Equation:
[tex]\[
2(x-2)^2 = 8(7 + y)
\][/tex]
2. Divide both sides by 8 to simplify:
[tex]\[
\frac{2(x-2)^2}{8} = 7 + y
\][/tex]
Simplifies to:
[tex]\[
\frac{(x-2)^2}{4} = 7 + y
\][/tex]
3. Subtract 7 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
\frac{(x-2)^2}{4} - 7 = y
\][/tex]
4. Simplify the expression:
[tex]\[
y = \frac{(x-2)^2}{4} - 7
\][/tex]
Now let's further expand and simplify the expression:
- Expand [tex]\( (x-2)^2 \)[/tex]:
[tex]\[
(x-2)^2 = x^2 - 4x + 4
\][/tex]
- Substitute back into the equation:
[tex]\[
y = \frac{x^2 - 4x + 4}{4} - 7
\][/tex]
- Distribute [tex]\( \frac{1}{4} \)[/tex] across the terms:
[tex]\[
y = \frac{x^2}{4} - x + 1 - 7
\][/tex]
- Combine like terms:
[tex]\[
y = \frac{x^2}{4} - x - 6
\][/tex]
Therefore, the inverse equation is:
[tex]\[
y = \frac{1}{4} x^2 - x - 6
\][/tex]
Among the given options, the inverse equation that matches our derived expression is:
[tex]\[
y = \frac{1}{4} x^2 - x - 6
\][/tex]
