Solving Algebraic Equations: Finding \(x^2 + \frac{1}{x^2}\)

We are given an algebraic equation and asked to find the value of a specific expression involving \(x\).

The given equation is: \(5x - \frac{5}{x} + 6 = 0\)

We need to find the value of: \(x^2 + \frac{1}{x^2}\)

Step-by-Step Solution for the Equation

Let's start by manipulating the given equation to isolate terms related to \(x\) and \(\frac{1}{x}\).

1. Rewrite the equation:

   \(5x - \frac{5}{x} + 6 = 0\)

2. Move the constant term to the right side of the equation:

   \(5x - \frac{5}{x} = -6\)

3. Factor out the common term, which is 5, from the left side:

   \(5\left(x - \frac{1}{x}\right) = -6\)

4. Divide both sides by 5 to isolate the expression \(\left(x - \frac{1}{x}\right)\):

   \(x - \frac{1}{x} = -\frac{6}{5}\)

Finding the Value of \(x^2 + \frac{1}{x^2}\)

We have the value of \(\left(x - \frac{1}{x}\right)\). We need to find the value of \(x^2 + \frac{1}{x^2}\). We can use the algebraic identity \((a-b)^2 = a^2 - 2ab + b^2\). Let \(a = x\) and \(b = \frac{1}{x}\).

Square both sides of the equation \(x - \frac{1}{x} = -\frac{6}{5}\):

\(\left(x - \frac{1}{x}\right)^2 = \left(-\frac{6}{5}\right)^2\)

Expand the left side using the identity:

\(x^2 - 2(x)\left(\frac{1}{x}\right) + \left(\frac{1}{x}\right)^2 = \left(-\frac{6}{5}\right)^2\)

Simplify the middle term and the right side:

\(x^2 - 2 + \frac{1}{x^2} = \frac{36}{25}\)

Now, isolate the term \(x^2 + \frac{1}{x^2}\) by adding 2 to both sides of the equation:

\(x^2 + \frac{1}{x^2} = \frac{36}{25} + 2\)

To add the terms on the right side, find a common denominator, which is 25:

\(x^2 + \frac{1}{x^2} = \frac{36}{25} + \frac{2 \times 25}{25}\)

\(x^2 + \frac{1}{x^2} = \frac{36}{25} + \frac{50}{25}\)

\(x^2 + \frac{1}{x^2} = \frac{36 + 50}{25}\)

\(x^2 + \frac{1}{x^2} = \frac{86}{25}\)

Comparison with Options

Let's compare our calculated value with the given options:

Our calculated value, \(\frac{86}{25}\), matches Option 1.

Algebraic Equation and Expression Analysis

The problem involves solving a rational algebraic equation and then evaluating a related algebraic expression. Key steps included rearranging the equation, factoring, and using a common algebraic identity.

Revision Table: Key Steps in Solving the Equation

Additional Information: Algebraic Identities

Algebraic identities are equalities that are true for all values of the variables involved. The identity used in this solution is \((a-b)^2 = a^2 - 2ab + b^2\). Another related identity is \((a+b)^2 = a^2 + 2ab + b^2\). These are derived from the distributive property of multiplication.

These identities are very useful in simplifying expressions and solving equations, especially when dealing with squares of sums or differences.

For example, if we had found \(\left(x + \frac{1}{x}\right)\) instead of \(\left(x - \frac{1}{x}\right)\), we would use the identity \((a+b)^2 = a^2 + 2ab + b^2\) to find \(x^2 + \frac{1}{x^2}\). Squaring \(\left(x + \frac{1}{x}\right)\) gives \(\left(x + \frac{1}{x}\right)^2 = x^2 + 2(x)\left(\frac{1}{x}\right) + \left(\frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2}\). From this, \(x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2\).

Similarly, from the identity \((a-b)^2 = a^2 - 2ab + b^2\), we can rearrange to get \(a^2 + b^2 = (a-b)^2 + 2ab\). In our case, with \(a=x\) and \(b=\frac{1}{x}\), we get \(x^2 + \left(\frac{1}{x}\right)^2 = \left(x - \frac{1}{x}\right)^2 + 2(x)\left(\frac{1}{x}\right)\), which simplifies to \(x^2 + \frac{1}{x^2} = \left(x - \frac{1}{x}\right)^2 + 2\). This confirms the step taken in the solution.