We are given the equation:

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

### Step 1: Rearrange the equation
Move all terms to one side of the equation:

\[
5x^4 - 9x^2 - 2 = 0
\]

### Step 2: Use substitution (optional)
To make the equation simpler, we can use substitution. Let \( y = x^2 \). Thus, the equation becomes:

\[
5y^2 - 9y - 2 = 0
\]

This is now a quadratic equation in terms of \( y \).

### Step 3: Solve the quadratic equation
We can solve this using the quadratic formula:

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

For our equation \( 5y^2 - 9y - 2 = 0 \), the coefficients are:
- \( a = 5 \)
- \( b = -9 \)
- \( c = -2 \)

Now, substitute these values into the quadratic formula:

\[
y = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(5)(-2)}}{2(5)}
\]

\[
y = \frac{9 \pm \sqrt{81 + 40}}{10}
\]

\[
y = \frac{9 \pm \sqrt{121}}{10}
\]

\[
y = \frac{9 \pm 11}{10}
\]

Thus, the two solutions for \( y \) are:

\[
y_1 = \frac{9 + 11}{10} = \frac{20}{10} = 2
\]
\[
y_2 = \frac{9 - 11}{10} = \frac{-2}{10} = -0.2
\]

### Step 4: Reverse substitution
Recall that \( y = x^2 \). Now substitute back to find \( x \):

For \( y = 2 \):

\[
x^2 = 2 \quad \Rightarrow \quad x = \pm \sqrt{2}
\]

For \( y = -0.2 \):

\[
x^2 = -0.2
\]

This gives no real solution, because the square of a real number cannot be negative.

### Final Solution:
The real solutions for \( x \) are:

\[
x = \pm \sqrt{2}
\]

Would you like more details or have any questions?

Here are some related questions you might consider:
1. How does the quadratic formula work for solving any quadratic equation?
2. Can negative values of \( y \) ever yield real solutions for \( x^2 \)?
3. How would you solve this equation graphically?
4. What are other methods to solve quadratic equations besides the quadratic formula?
5. How does substitution simplify solving higher-degree polynomials?

**Tip:** Always check the domain of your solution when dealing with square roots, to ensure real or complex solutions.

Explore how to solve the polynomial equation 9x^2 + 2 = 5x^4 using the substitution method and the quadratic formula. This detailed step-by-step guide breaks down the process of rearranging the equation, applying substitution, and solving for the real values of x. Suitable for 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