solve the equation 
𝑥
2
+
9
𝑥
2
=
10
x 
2
 + 
x 
2
 
9
​
 =10

The given equation is:

\[
x^2 + \frac{9}{x^2} = 10
\]

Let's solve it step by step.

### Step 1: Let \( y = x^2 \)

This substitution simplifies the equation:

\[
y + \frac{9}{y} = 10
\]

### Step 2: Multiply the equation by \( y \) to eliminate the fraction

\[
y^2 + 9 = 10y
\]

Rearrange this equation:

\[
y^2 - 10y + 9 = 0
\]

### Step 3: Solve the quadratic equation

To solve \( y^2 - 10y + 9 = 0 \), we use the quadratic formula:

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

where \( a = 1 \), \( b = -10 \), and \( c = 9 \). Plugging these values in:

\[
y = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}
\]

\[
y = \frac{10 \pm \sqrt{100 - 36}}{2}
\]

\[
y = \frac{10 \pm \sqrt{64}}{2}
\]

\[
y = \frac{10 \pm 8}{2}
\]

This gives two solutions:

\[
y = \frac{18}{2} = 9 \quad \text{and} \quad y = \frac{2}{2} = 1
\]

### Step 4: Find \( x \) from \( y = x^2 \)

Since \( y = 9 \) and \( y = 1 \), we have:

1. \( x^2 = 9 \) gives \( x = \pm 3 \)
2. \( x^2 = 1 \) gives \( x = \pm 1 \)

### Final Solutions

The solutions for the original equation are:

\[
x = \pm 3, \pm 1
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you solve quadratic equations using factorization?
2. Can you explain the discriminant in the quadratic formula and its significance?
3. How does substituting variables simplify solving equations?
4. What are other methods to solve equations involving fractions?
5. Can quadratic equations have complex solutions?

### Tip:
Always check your solutions by substituting them back into the original equation to ensure they satisfy it.

Solve the equation x^2 + 9/x^2 = 10 step-by-step using substitution and the quadratic formula. This problem demonstrates key algebraic techniques including solving quadratic equations and working with fractional expressions. Suitable for students in Grades 9-11.

Math Problem Statement

solve the equation 𝑥 2 + 9 𝑥 2

Solution

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