Which equation can be simplified to find the inverse of y = 5x2 + 10?
x = 5y2 + 10
StartFraction 1 Over y EndFraction = 5 x squared + 10
–y = 5x2 + 10
y = one-fifth x squared + one-tenth

To find the inverse of the equation \( y = 5x^2 + 10 \), the first step is to rearrange it so that \( x \) is expressed in terms of \( y \). Here's how the process works:

1. Start with the equation:
   \[
   y = 5x^2 + 10
   \]
2. Subtract 10 from both sides:
   \[
   y - 10 = 5x^2
   \]
3. Divide both sides by 5:
   \[
   \frac{y - 10}{5} = x^2
   \]
4. Take the square root of both sides to solve for \( x \):
   \[
   x = \pm \sqrt{\frac{y - 10}{5}}
   \]

Thus, the inverse of the equation is \( x = \pm \sqrt{\frac{y - 10}{5}} \).

Now, analyzing the given options:

1. **\( x = 5y^2 + 10 \)**: This does not represent the inverse; it's a rearranged form but does not help isolate \( x \) correctly.
2. **\( \frac{1}{y} = 5x^2 + 10 \)**: This equation isn't related to finding the inverse.
3. **\( -y = 5x^2 + 10 \)**: Incorrect, as multiplying both sides by \(-1\) doesn't lead to the correct inverse.
4. **\( y = \frac{1}{5}x^2 + \frac{1}{10} \)**: This is a simpler form of the original equation, not its inverse.

None of the provided options exactly represent the inverse, but the closest form that could be simplified is **\( y = \frac{1}{5}x^2 + \frac{1}{10} \)**, though it's not the inverse directly.

Would you like further details or have any questions?

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Here are 5 related questions to expand on this:

1. How can you verify if a function is invertible?
2. What is the significance of the square root in finding the inverse?
3. Why are there two possible values for \( x \) in the inverse?
4. How can we graphically interpret the inverse of a quadratic function?
5. What is the domain of the inverse function?

**Tip**: Always check the domain and range of the original function before finding its inverse!

Which equation can be simplified to find the inverse of y = 5x^2 + 10?

Explore how to find the inverse of the quadratic function y = 5x^2 + 10 with a step-by-step solution. This problem involves key algebraic concepts like inverse functions and quadratic equations, suitable for Grades 9-11.

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