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Which equation is the inverse of [tex]\(2(x-2)^2=8(7+y)\)[/tex]?

A. [tex]\(-2(x-2)^2=-8(7+y)\)[/tex]
B. [tex]\(y=\frac{1}{4} x^2-x-6\)[/tex]
C. [tex]\(y=-2 \pm \sqrt{28+4 x}\)[/tex]
D. [tex]\(y=2 \pm \sqrt{28+4 x}\)[/tex]



Answer :

GinnyAnswer
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]

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