To find the inverse of the given equation [tex]\( 2(x - 2)^2 = 8(7 + y) \)[/tex], we'll follow these steps:
1. Start with the original equation:
[tex]\[ 2(x - 2)^2 = 8(7 + y) \][/tex]
2. Divide both sides by 2 to simplify:
[tex]\[ (x - 2)^2 = 4(7 + y) \][/tex]
3. Isolate the term containing y by taking the square root of both sides:
[tex]\[ x - 2 = \pm 2\sqrt{7 + y} \][/tex]
4. Solve for [tex]\( y \)[/tex] by isolating it:
[tex]\[ \frac{x - 2}{2} = \pm \sqrt{7 + y} \][/tex]
5. Square both sides to eliminate the square root:
[tex]\[ \left( \frac{x - 2}{2} \right)^2 = 7 + y \][/tex]
6. Simplify the squared term on the left side:
[tex]\[ \frac{(x - 2)^2}{4} = 7 + y \][/tex]
7. Isolate [tex]\( y \)[/tex] by subtracting 7 from both sides:
[tex]\[ y = \frac{(x - 2)^2}{4} - 7 \][/tex]
Thus, the inverse equation is:
[tex]\[ y = \frac{(x - 2)^2}{4} - 7 \][/tex]
So, the correct inverse equation from the given options is:
[tex]\[ y = \frac{(x - 2)^2}{4} - 7 \][/tex]
