To solve the equation \( 2^y - x^2 = 615 \), we need to find values for \( x \) and \( y \) that satisfy this equation.

We can rewrite the equation as:

\[
2^y = x^2 + 615
\]

### Method: Trial and error (since \( y \) and \( x \) are often integers)

#### Step 1: Try different integer values for \( x \).

- If \( x = 25 \), then:
  \[
  2^y = 25^2 + 615 = 625 + 615 = 1240
  \]
  Solving \( 2^y = 1240 \), but this doesn't yield an integer \( y \).

- If \( x = 24 \), then:
  \[
  2^y = 24^2 + 615 = 576 + 615 = 1191
  \]
  \( 2^y = 1191 \) is also not a valid power of 2.

- If \( x = 23 \), then:
  \[
  2^y = 23^2 + 615 = 529 + 615 = 1144
  \]
  \( 2^y = 1144 \) is not valid either.

- **If \( x = 21 \)**, then:
  \[
  2^y = 21^2 + 615 = 441 + 615 = 1056
  \]
  \( 2^y = 1056 \), but this is also not a valid power of 2.

- **If \( x = 17 \)**:
  \[
  2^y = 17^2 + 615 = 289 + 615 = 904
  \]
  \( 2^y = 904 \) also doesn't work.

You can continue trial and error until a valid combination is found, or we can shift to a different approach.

Would you like more trial combinations, or another method like graphing or constraints?

Here are 5 questions to extend this:

1. What happens if \( y \) is constrained to even numbers only?
2. Can you find another equation where powers of 2 are related to quadratic functions?
3. How do you graph exponential vs quadratic equations?
4. How can logarithms help in exponential equations like this?
5. Why are trial-and-error methods useful in some nonlinear equations?

**Tip:** Sometimes it's useful to isolate one variable and try plotting or tabulating values to spot patterns.

Discover how to solve the equation 2^y - x^2 = 615 through trial and error. This problem demonstrates the interaction between exponential and quadratic functions, and the solution explores integer values for x and y. Suitable for students in Grades 9-12.

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