To find the inverse of the equation [tex]\( y = 5x^2 + 10 \)[/tex], we need to follow a step-by-step approach to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] and then express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] again.
1. Start with the given equation:
[tex]\[
y = 5x^2 + 10
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the equation, which represents the inverse relationship:
[tex]\[
x = 5y^2 + 10
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
a. Subtract 10 from both sides:
[tex]\[
x - 10 = 5y^2
\][/tex]
b. Divide both sides by 5:
[tex]\[
\frac{x - 10}{5} = y^2
\][/tex]
c. Take the square root of both sides:
[tex]\[
y = \pm \sqrt{\frac{x - 10}{5}}
\][/tex]
Therefore, the simplified equation to find the inverse is:
[tex]\[
x = 5y^2 + 10 \implies y = \pm \sqrt{\frac{x - 10}{5}}
\][/tex]
Among the provided choices, the equation that can be simplified to find the inverse is:
[tex]\[
x = 5y^2 + 10
\][/tex]
