Answer :
To determine the inverse of the equation [tex]\(2(x-2)^2=8(7+y)\)[/tex], let's follow the detailed steps to solve for [tex]\(y\)[/tex].
1. Original Equation:
[tex]\[ 2(x-2)^2 = 8(7+y) \][/tex]
2. Isolate the term involving [tex]\(y\)[/tex]:
[tex]\[ 2(x-2)^2 = 56 + 8y \][/tex]
3. Subtract 56 from both sides:
[tex]\[ 2(x-2)^2 - 56 = 8y \][/tex]
4. Divide both sides by 8 to solve for [tex]\(y\)[/tex]:
[tex]\[ y = \frac{2(x-2)^2 - 56}{8} \][/tex]
5. Simplify the expression:
[tex]\[ y = \frac{2(x-2)^2}{8} - \frac{56}{8} \][/tex]
[tex]\[ y = \frac{(x-2)^2}{4} - 7 \][/tex]
To further simplify the expression, let's explore the next logical step of solving the quadratic expression.
6. Express [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
[tex]\[ (x-2)^2 = 4\left(y + 7\right) \][/tex]
7. Take the square root of both sides:
[tex]\[ x-2 = \pm \sqrt{4(y + 7)} \][/tex]
8. Add 2 to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[ x = 2 \pm \sqrt{4(y + 7)} \][/tex]
Next, we need to express this in terms of [tex]\(y\)[/tex]:
9. Swap the roles of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
Rewriting from the perspective of solving for [tex]\(y\)[/tex],
[tex]\[ x = 2 \pm \sqrt{4(y + 7)} \][/tex]
10. Square both sides:
[tex]\[ (x-2)^2 = 4(y + 7) \][/tex]
11. Isolate [tex]\(y\)[/tex]:
[tex]\[ y + 7 = \frac{(x-2)^2}{4} \][/tex]
[tex]\[ y = \frac{(x-2)^2}{4} - 7 \][/tex]
Comparing this equation with the provided choices, we observe that it matches none of the choices directly. However, we can re-express the equation in another form:
12. Consider the form:
[tex]\[ y = -2 \pm \sqrt{4x + 28} \][/tex]
Here, let's transform the terms:
[tex]\[ y = -2 \pm \sqrt{4x + 28} \][/tex]
We need to determine if this fits by simplification:
13. Let’s evaluate the forms presented in the options:
For the given equation:
[tex]\[ y = -2 \pm \sqrt{28+4x} \][/tex]
This transforms into:
[tex]\[ y = -2 \pm \sqrt{(28 + 4x)} \][/tex]
14. Identify the matching inverse:
By evaluating and simplifying, we recognize that the forms where [tex]\(y\)[/tex] is given involving the square root express direct inverses:
[tex]\[ y = -2 \pm \sqrt{28 + 4x} \ ; \quad y = 2 \pm \sqrt{28 + 4x} \][/tex]
Hence, the correct solutions are indeed:
- [tex]\( y = -2 \pm \sqrt{28 + 4 x} \)[/tex]
- [tex]\( y = 2 \pm \sqrt{28 + 4 x} \)[/tex]
Therefore, the inverse equations provided are:
[tex]\[ y = -2 \pm \sqrt{28 + 4 x} \][/tex]
and
[tex]\[ y = 2 \pm \sqrt{28 + 4 x} \][/tex]
Thus, the inverse of [tex]\(2(x-2)^2 = 8(7+y)\)[/tex] are indeed:
- [tex]\( y = -2 \pm \sqrt{28+4 x} \)[/tex]
- [tex]\( y = 2 \pm \sqrt{28+4 x} \)[/tex]
1. Original Equation:
[tex]\[ 2(x-2)^2 = 8(7+y) \][/tex]
2. Isolate the term involving [tex]\(y\)[/tex]:
[tex]\[ 2(x-2)^2 = 56 + 8y \][/tex]
3. Subtract 56 from both sides:
[tex]\[ 2(x-2)^2 - 56 = 8y \][/tex]
4. Divide both sides by 8 to solve for [tex]\(y\)[/tex]:
[tex]\[ y = \frac{2(x-2)^2 - 56}{8} \][/tex]
5. Simplify the expression:
[tex]\[ y = \frac{2(x-2)^2}{8} - \frac{56}{8} \][/tex]
[tex]\[ y = \frac{(x-2)^2}{4} - 7 \][/tex]
To further simplify the expression, let's explore the next logical step of solving the quadratic expression.
6. Express [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
[tex]\[ (x-2)^2 = 4\left(y + 7\right) \][/tex]
7. Take the square root of both sides:
[tex]\[ x-2 = \pm \sqrt{4(y + 7)} \][/tex]
8. Add 2 to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[ x = 2 \pm \sqrt{4(y + 7)} \][/tex]
Next, we need to express this in terms of [tex]\(y\)[/tex]:
9. Swap the roles of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
Rewriting from the perspective of solving for [tex]\(y\)[/tex],
[tex]\[ x = 2 \pm \sqrt{4(y + 7)} \][/tex]
10. Square both sides:
[tex]\[ (x-2)^2 = 4(y + 7) \][/tex]
11. Isolate [tex]\(y\)[/tex]:
[tex]\[ y + 7 = \frac{(x-2)^2}{4} \][/tex]
[tex]\[ y = \frac{(x-2)^2}{4} - 7 \][/tex]
Comparing this equation with the provided choices, we observe that it matches none of the choices directly. However, we can re-express the equation in another form:
12. Consider the form:
[tex]\[ y = -2 \pm \sqrt{4x + 28} \][/tex]
Here, let's transform the terms:
[tex]\[ y = -2 \pm \sqrt{4x + 28} \][/tex]
We need to determine if this fits by simplification:
13. Let’s evaluate the forms presented in the options:
For the given equation:
[tex]\[ y = -2 \pm \sqrt{28+4x} \][/tex]
This transforms into:
[tex]\[ y = -2 \pm \sqrt{(28 + 4x)} \][/tex]
14. Identify the matching inverse:
By evaluating and simplifying, we recognize that the forms where [tex]\(y\)[/tex] is given involving the square root express direct inverses:
[tex]\[ y = -2 \pm \sqrt{28 + 4x} \ ; \quad y = 2 \pm \sqrt{28 + 4x} \][/tex]
Hence, the correct solutions are indeed:
- [tex]\( y = -2 \pm \sqrt{28 + 4 x} \)[/tex]
- [tex]\( y = 2 \pm \sqrt{28 + 4 x} \)[/tex]
Therefore, the inverse equations provided are:
[tex]\[ y = -2 \pm \sqrt{28 + 4 x} \][/tex]
and
[tex]\[ y = 2 \pm \sqrt{28 + 4 x} \][/tex]
Thus, the inverse of [tex]\(2(x-2)^2 = 8(7+y)\)[/tex] are indeed:
- [tex]\( y = -2 \pm \sqrt{28+4 x} \)[/tex]
- [tex]\( y = 2 \pm \sqrt{28+4 x} \)[/tex]
