If `(x^(2)-1)/(x)=sqrt5` and x is positive number find `(x^(2) + (1)/(x^(2))) (x+ (1)/(x))=` ?

Answer

To solve the problem step by step, we start with the equation given:

\[ \frac{x^2 - 1}{x} = \sqrt{5} \]

### Step 1: Rewrite the equation
We can rewrite the equation as follows:

\[ x - \frac{1}{x} = \sqrt{5} \]

**Hint:** Rearranging the equation can help simplify the problem.

### Step 2: Square both sides
Next, we square both sides of the equation:

\[ \left(x - \frac{1}{x}\right)^2 = (\sqrt{5})^2 \]

This gives us:

\[ x^2 - 2\cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 5 \]

**Hint:** Remember that \((a - b)^2 = a^2 - 2ab + b^2\).

### Step 3: Simplify the equation
Now, simplify the equation:

\[ x^2 - 2 + \frac{1}{x^2} = 5 \]

Adding 2 to both sides results in:

\[ x^2 + \frac{1}{x^2} = 7 \]

**Hint:** Isolate the term you want to find by moving other terms to the other side.

### Step 4: Find \(x + \frac{1}{x}\)
Now, we need to find \(x + \frac{1}{x}\). We can use the identity:

\[ \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} \]

From the previous step, we know:

\[ x^2 + \frac{1}{x^2} = 7 \]

So we can substitute:

\[ \left(x + \frac{1}{x}\right)^2 = 7 + 2 = 9 \]

Taking the square root of both sides gives:

\[ x + \frac{1}{x} = 3 \]

**Hint:** Use the square root to find the positive solution since \(x\) is positive.

### Step 5: Calculate the final expression
Now we need to find:

\[ \left(x^2 + \frac{1}{x^2}\right) \left(x + \frac{1}{x}\right) \]

Substituting the values we found:

\[ 7 \cdot 3 = 21 \]

### Final Answer
Thus, the final answer is:

\[ \boxed{21} \]

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