We need to solve the equation:

\[
-x^4 + x^2 + 2 = 0
\]

### Step 1: Substitution
To simplify the process, we use the substitution:
\[
y = x^2
\]
This transforms the equation into:
\[
-y^2 + y + 2 = 0
\]
which is a quadratic equation in \( y \).

### Step 2: Solve the quadratic equation
We solve this quadratic equation using the quadratic formula:
\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( -y^2 + y + 2 = 0 \), we have:
\[
a = -1, \quad b = 1, \quad c = 2
\]

Substitute these values into the quadratic formula:
\[
y = \frac{-1 \pm \sqrt{1^2 - 4(-1)(2)}}{2(-1)}
\]
\[
y = \frac{-1 \pm \sqrt{1 + 8}}{-2}
\]
\[
y = \frac{-1 \pm \sqrt{9}}{-2}
\]
\[
y = \frac{-1 \pm 3}{-2}
\]

### Step 3: Find the roots for \( y \)
Now solve for \( y \):
1. \( y = \frac{-1 + 3}{-2} = \frac{2}{-2} = -1 \)
2. \( y = \frac{-1 - 3}{-2} = \frac{-4}{-2} = 2 \)

Thus, \( y = -1 \) and \( y = 2 \).

### Step 4: Back-substitute \( y = x^2 \)
Since \( y = x^2 \), we solve for \( x \):
1. If \( y = -1 \), there is no real solution for \( x \) because \( x^2 = -1 \) has no real roots.
2. If \( y = 2 \), then \( x^2 = 2 \), so:
   \[
   x = \pm \sqrt{2}
   \]

### Final Answer:
The real solutions are:
\[
x = \pm \sqrt{2}
\]

Would you like more details on any of these steps?

Here are some related questions you might find interesting:
1. What happens if the equation has complex roots?
2. How does the quadratic formula work for solving any quadratic equation?
3. Can we apply this method to higher-degree polynomials?
4. How do you check if an equation has real or complex solutions?
5. What are other ways to solve quartic equations?

**Tip:** Always check for possible complex solutions, especially when you get negative results inside square roots.

This detailed solution walks through the steps of solving the quartic equation -x^4 + x^2 + 2 = 0 by using substitution. Learn key algebraic techniques such as the quadratic formula and substitution, and explore the process of finding real and complex solutions. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation