https://file.aiquickdraw.com/mathbot/file/17265303335108g8ysedg.jpg

The equation given is:

\[
(x^2 - 5)^2 = 16
\]

To solve for \( x \), let's first take the square root of both sides of the equation.

\[
|x^2 - 5| = 4
\]

This absolute value equation can now be broken into two cases:

### Case 1: 
\[
x^2 - 5 = 4
\]

Solve for \( x^2 \):

\[
x^2 = 4 + 5 = 9
\]

So, \( x = \pm 3 \). This gives two solutions: \( x = 3 \) and \( x = -3 \).

### Case 2:
\[
x^2 - 5 = -4
\]

Solve for \( x^2 \):

\[
x^2 = -4 + 5 = 1
\]

So, \( x = \pm 1 \). This gives two more solutions: \( x = 1 \) and \( x = -1 \).

### Conclusion:

In total, there are **four different real numbers** that satisfy the equation: \( x = 3, -3, 1, -1 \).

Thus, the correct answer is **(D) 4**.

Would you like further details or have any questions?

---

Here are 5 related questions you might find helpful:
1. What is the method for solving absolute value equations in general?
2. How do you verify the solutions of this equation?
3. Could this type of problem be solved graphically?
4. What would happen if the equation involved complex numbers instead?
5. How do we approach equations that involve higher powers, like \( (x^2 - 5)^4 \)?

**Tip**: Always remember to check all possible solutions when dealing with absolute value equations to avoid missing any solutions!

How many different real numbers x satisfy the equation (x^2 - 5)^2 = 16?

Learn how to solve the equation (x^2 - 5)^2 = 16 with a step-by-step breakdown. This problem covers key algebraic concepts including absolute value equations and quadratic equations. Suitable for high school students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation