Understanding the Algebraic Problem

The question asks us to find the value of a specific algebraic expression, \(\frac{x^4 + \frac{1}{x^2}}{x^2 + 1}\), given an initial equation involving square roots: \(\sqrt{x} - \frac{1}{\sqrt{x}} = \sqrt{5}\). We are also given that \(x \ne 0\).

To solve this, we first need to use the given equation to find a simpler relationship involving \(x\), and then substitute or manipulate the target expression to use this relationship.

Step-by-Step Solution for the Expression Value

Step 1: Simplify the Initial Equation \(\sqrt{x} - \frac{1}{\sqrt{x}} = \sqrt{5}\)

We are given the equation \(\sqrt{x} - \frac{1}{\sqrt{x}} = \sqrt{5}\). To eliminate the square roots, we can square both sides of the equation.

Squaring both sides:

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

Using the identity \((a-b)^2 = a^2 - 2ab + b^2\), where \(a = \sqrt{x}\) and \(b = \frac{1}{\sqrt{x}}\):

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

\(x - 2(1) + \frac{1}{x} = 5\)

\(x - 2 + \frac{1}{x} = 5\)

Add 2 to both sides:

\(x + \frac{1}{x} = 5 + 2\)

\(x + \frac{1}{x} = 7\)

This gives us a fundamental relationship between \(x\) and \(\frac{1}{x}\).

Step 2: Manipulate the Target Expression \(\frac{x^4 + \frac{1}{x^2}}{x^2 + 1}\)

The expression we need to evaluate is \(\frac{x^4 + \frac{1}{x^2}}{x^2 + 1}\). This expression looks a bit unusual with the mix of powers in the numerator. Let's try to simplify it by multiplying the numerator and the denominator by \(x^2\).

\(\frac{\left(x^4 + \frac{1}{x^2}\right) \times x^2}{(x^2 + 1) \times x^2}\)

Distribute \(x^2\) in the numerator:

\(\frac{x^4 \times x^2 + \frac{1}{x^2} \times x^2}{x^2(x^2 + 1)}\)

\(\frac{x^6 + 1}{x^2(x^2 + 1)}\)

Step 3: Use the Relationship \(x + \frac{1}{x} = 7\) to Simplify Parts of the Expression

From Step 1, we have \(x + \frac{1}{x} = 7\). Let's see if we can use this to simplify the denominator \(x^2(x^2 + 1)\).

Multiply the equation \(x + \frac{1}{x} = 7\) by \(x\) (since \(x \ne 0\)):

\(x\left(x + \frac{1}{x}\right) = 7x\)

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

\(x^2 + 1 = 7x\)

Now substitute \(x^2 + 1 = 7x\) into the denominator of the expression from Step 2:

\(\frac{x^6 + 1}{x^2(x^2 + 1)} = \frac{x^6 + 1}{x^2(7x)} = \frac{x^6 + 1}{7x^3}\)

We can rewrite this as:

\(\frac{1}{7} \times \frac{x^6 + 1}{x^3}\)

Now, let's simplify the fraction \(\frac{x^6 + 1}{x^3}\):

\(\frac{x^6}{x^3} + \frac{1}{x^3} = x^3 + \frac{1}{x^3}\)

So the expression becomes \(\frac{1}{7} \times \left(x^3 + \frac{1}{x^3}\right)\).

Step 4: Calculate the Value of \(x^3 + \frac{1}{x^3}\)

We know that \(x + \frac{1}{x} = 7\). We need to find \(x^3 + \frac{1}{x^3}\). We can use the algebraic identity \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\) or \(a^3 + b^3 = (a+b)^3 - 3ab(a+b)\). Using \(a=x\) and \(b=\frac{1}{x}\):

\(x^3 + \frac{1}{x^3} = \left(x + \frac{1}{x}\right)^3 - 3\left(x\right)\left(\frac{1}{x}\right)\left(x + \frac{1}{x}\right)\)

Substitute the value \(x + \frac{1}{x} = 7\):

\(x^3 + \frac{1}{x^3} = (7)^3 - 3(1)(7)\)

\(x^3 + \frac{1}{x^3} = 343 - 21\)

\(x^3 + \frac{1}{x^3} = 322\)

Step 5: Substitute and Calculate the Final Value

Now substitute the value of \(x^3 + \frac{1}{x^3}\) back into the expression from Step 3, which was \(\frac{1}{7} \times \left(x^3 + \frac{1}{x^3}\right)\).

Value of the expression = \(\frac{1}{7} \times (322)\)

Value of the expression = \(\frac{322}{7}\)

Performing the division:

\(322 \div 7 = 46\)

Final Answer for the Algebraic Expression

The value of the expression \(\frac{x^4 + \frac{1}{x^2}}{x^2 + 1}\) is 46.

Revision Table: Key Algebraic Identities Used

Additional Information on Solving Algebraic Problems

When faced with algebraic problems involving roots, fractions, and powers, simplifying the initial conditions is often the first step. Squaring both sides of an equation with square roots is a common technique to eliminate the roots.

Recognizing common patterns like \(x + \frac{1}{x}\), \(x^2 + \frac{1}{x^2}\), and \(x^3 + \frac{1}{x^3}\) and knowing their relationships is very helpful. If \(x + \frac{1}{x} = k\), then:

• \(x^2 + \frac{1}{x^2} = (x + \frac{1}{x})^2 - 2 = k^2 - 2\)
• \(x^3 + \frac{1}{x^3} = (x + \frac{1}{x})^3 - 3(x + \frac{1}{x}) = k^3 - 3k\)

Manipulating complex expressions by multiplying the numerator and denominator by a suitable term (like \(x^2\) in this case) can often reveal simpler structures that can be evaluated using the derived relationships.

Always pay attention to any restrictions given, like \(x \ne 0\), as these ensure validity of operations like division by \(x\) or multiplication by \(x\).