2. If $$ x+\frac{1}{x}=9 $$, find $$ x^{2}+\frac{1}{x^{2}} $$

Question

UpStudy AI · Accepted Answer

To find $$ x^{2} + \frac{1}{x^{2}} $$ given that $$ x + \frac{1}{x} = 9 $$, follow these steps:

1. **Square Both Sides of the Given Equation:**

$$
   \left( x + \frac{1}{x} \right)^2 = 9^2
   $$

$$
   x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 81
   $$

2. **Simplify the Equation:**

Notice that $$ 2 \cdot x \cdot \frac{1}{x} = 2 $$. So,

$$
   x^2 + 2 + \frac{1}{x^2} = 81
   $$

3. **Isolate $$ x^{2} + \frac{1}{x^{2}} $$:**

$$
   x^2 + \frac{1}{x^2} = 81 - 2
   $$

$$
   x^2 + \frac{1}{x^2} = 79
   $$

**Answer:** $$ x^{2} + \frac{1}{x^{2}} = 79 $$
$$ x^{2} + \frac{1}{x^{2}} = 79 $$

2. If \( x+\frac{1}{x}=9 \), find \( x^{2}+\frac{1}{x^{2}} \)