If ` 2 x + ((9)/( x)) = 9 ` what is te minimum value of ` x^(2) + ((1)/( x^(2))) ` ?

Options

A. `(95)/(36)`
B. `(97)/(36)`
C. `(86)/(25)`
D. `(623)/(27)`

Answer

To solve the equation \( 2x + \frac{9}{x} = 9 \) and find the minimum value of \( x^2 + \frac{1}{x^2} \), we can follow these steps:

### Step 1: Rearrange the equation
Start with the given equation:
\[ 2x + \frac{9}{x} = 9 \]
Subtract \( 9 \) from both sides:
\[ 2x + \frac{9}{x} - 9 = 0 \]
This simplifies to:
\[ 2x + \frac{9}{x} - 9 = 0 \]

### Step 2: Multiply through by \( x \) to eliminate the fraction
To get rid of the fraction, multiply the entire equation by \( x \) (assuming \( x \neq 0 \)):
\[ 2x^2 + 9 - 9x = 0 \]
Rearranging gives:
\[ 2x^2 - 9x + 9 = 0 \]

### Step 3: Solve the quadratic equation
Now we can solve the quadratic equation \( 2x^2 - 9x + 9 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 2 \), \( b = -9 \), and \( c = 9 \):
\[ x = \frac{9 \pm \sqrt{(-9)^2 - 4 \cdot 2 \cdot 9}}{2 \cdot 2} \]
Calculating the discriminant:
\[ x = \frac{9 \pm \sqrt{81 - 72}}{4} = \frac{9 \pm \sqrt{9}}{4} = \frac{9 \pm 3}{4} \]
This gives us two solutions:
\[ x = \frac{12}{4} = 3 \quad \text{and} \quad x = \frac{6}{4} = \frac{3}{2} \]

### Step 4: Calculate \( x^2 + \frac{1}{x^2} \) for both values of \( x \)
Now we will find \( x^2 + \frac{1}{x^2} \) for both values of \( x \).

1. For \( x = 3 \):
\[ x^2 + \frac{1}{x^2} = 3^2 + \frac{1}{3^2} = 9 + \frac{1}{9} = 9 + 0.1111 = \frac{81 + 1}{9} = \frac{82}{9} \approx 9.1111 \]

2. For \( x = \frac{3}{2} \):
\[ x^2 + \frac{1}{x^2} = \left(\frac{3}{2}\right)^2 + \frac{1}{\left(\frac{3}{2}\right)^2} = \frac{9}{4} + \frac{1}{\frac{9}{4}} = \frac{9}{4} + \frac{4}{9} \]
Finding a common denominator (36):
\[ = \frac{81}{36} + \frac{16}{36} = \frac{97}{36} \approx 2.6944 \]

### Step 5: Compare the values
Now we compare the two results:
- For \( x = 3 \), \( x^2 + \frac{1}{x^2} \approx 9.1111 \)
- For \( x = \frac{3}{2} \), \( x^2 + \frac{1}{x^2} \approx 2.6944 \)

### Conclusion
The minimum value of \( x^2 + \frac{1}{x^2} \) is:
\[ \frac{97}{36} \]