Understanding the Problem: Solving for \(x^2 + \frac{1}{x^2}\)

The question asks for the value of the expression \(x^2 + \frac{1}{x^2}\) given the quadratic equation \(x^2 - 8x - 1 = 0\). We need to use the given equation to find a relationship involving \(x\) and \(\frac{1}{x}\) and then use that relationship to calculate the required expression.

Step-by-Step Solution

We are given the equation:

\[x^2 - 8x - 1 = 0\]

Our goal is to find the value of \(x^2 + \frac{1}{x^2}\). Let's try to manipulate the given equation to get terms involving \(x\) and \(\frac{1}{x}\).

First, observe that if \(x=0\), the equation becomes \(0^2 - 8(0) - 1 = 0\), which simplifies to \(-1 = 0\). This is false, so \(x\) cannot be 0. Since \(x \ne 0\), we can divide the entire equation by \(x\).

Dividing the equation \(x^2 - 8x - 1 = 0\) by \(x\):

\[ \frac{x^2}{x} - \frac{8x}{x} - \frac{1}{x} = \frac{0}{x} \] \[ x - 8 - \frac{1}{x} = 0 \]

Now, rearrange this equation to isolate terms involving \(x\) and \(\frac{1}{x}\):

\[ x - \frac{1}{x} = 8 \]

This gives us a useful relationship between \(x\) and \(\frac{1}{x}\). Now, consider the expression we need to find: \(x^2 + \frac{1}{x^2}\).

Recall the algebraic identity for squaring a difference: \((a - b)^2 = a^2 - 2ab + b^2\). Let \(a = x\) and \(b = \frac{1}{x}\).

So, we can square the expression \(x - \frac{1}{x}\):

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

We already found that \(x - \frac{1}{x} = 8\). Substitute this value into the squared expression:

\[ (8)^2 = x^2 - 2 + \frac{1}{x^2} \] \[ 64 = x^2 - 2 + \frac{1}{x^2} \]

Now, we can rearrange this equation to solve for \(x^2 + \frac{1}{x^2}\):

\[ 64 + 2 = x^2 + \frac{1}{x^2} \] \[ 66 = x^2 + \frac{1}{x^2} \]

Therefore, the value of \(x^2 + \frac{1}{x^2}\) is 66.

Checking the Options

Let's look at the given options:

1. 68
2. 62
3. 64
4. 66

Our calculated value for \(x^2 + \frac{1}{x^2}\) is 66, which matches option 4.

Key Concepts Used

• Solving algebraic equations by manipulating terms.
• Dividing an equation by a variable (after ensuring the variable is not zero).
• Using algebraic identities, specifically \((a - b)^2 = a^2 - 2ab + b^2\), to relate expressions.

Revision Table: Quadratic Equation and Expressions

Additional Information: Alternative Identity Approach

Another useful identity is \((a+b)^2 = a^2 + 2ab + b^2\). If we were given an equation that led to \(x + \frac{1}{x} = k\), we could square it:

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

So, \(x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2\). This shows that the value of \(x^2 + \frac{1}{x^2}\) can be derived from either \(x - \frac{1}{x}\) or \(x + \frac{1}{x}\) using appropriate algebraic manipulation.

In our problem, dividing \(x^2 - 8x - 1 = 0\) by \(x\) naturally led to \(x - \frac{1}{x} = 8\), making the \((a-b)^2\) identity the most direct path to the solution.