The equation you've provided is:

\[
x^4 - 24x^2 - 25 = 0
\]

This is a quadratic equation in terms of \(x^2\). To solve it, we can perform a substitution. Let's set \( y = x^2 \). The equation then becomes:

\[
y^2 - 24y - 25 = 0
\]

Now, we'll solve this quadratic equation for \(y\) using the quadratic formula:

\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(y^2 - 24y - 25 = 0\):

- \(a = 1\)
- \(b = -24\)
- \(c = -25\)

Substituting these values into the quadratic formula gives:

\[
y = \frac{24 \pm \sqrt{(-24)^2 - 4(1)(-25)}}{2(1)}
\]
\[
y = \frac{24 \pm \sqrt{576 + 100}}{2}
\]
\[
y = \frac{24 \pm \sqrt{676}}{2}
\]
\[
y = \frac{24 \pm 26}{2}
\]

This gives us two possible values for \(y\):

1. \(y = \frac{24 + 26}{2} = \frac{50}{2} = 25\)
2. \(y = \frac{24 - 26}{2} = \frac{-2}{2} = -1\)

Since \(y = x^2\), \(x^2 = 25\) or \(x^2 = -1\).

- For \(x^2 = 25\), \(x = \pm 5\).
- For \(x^2 = -1\), there are no real solutions since the square of a real number cannot be negative. However, in the complex number system, the solutions would be \(x = \pm i\).

Thus, the solutions to the original equation are:

\[
x = 5, \, -5, \, i, \, -i
\]

Would you like more details or have any questions about this solution?

Here are some related questions:
1. How does the quadratic formula work?
2. What is the significance of the discriminant in a quadratic equation?
3. How do you solve equations involving complex numbers?
4. Can every polynomial be factored into real numbers?
5. How can substitution help in solving higher-degree equations?

**Tip:** Remember that if you encounter a negative number under a square root while solving equations, the solutions will involve complex numbers.

Learn how to solve the quadratic equation x^4 - 24x^2 - 25 = 0 with detailed step-by-step instructions. Explore the application of the quadratic formula and complex numbers in solving higher-degree equations. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation