Problem 30 - A Graphical Approach to Precalculus with Limits

Question

Find all complex solutions for each equation by hand. $$x(x-2)^{-1}+x(x+2)^{-1}=2\left(x^{2}-4\right)^{-1}$$

Step-by-Step Solution

Answer

The solutions are \(x = 1\) and \(x = -1\).

Step 1: Understand the Equation

The original equation is given as \(x(x-2)^{-1} + x(x+2)^{-1} = 2(x^2-4)^{-1}\). This can be rewritten using fraction notation: \(\frac{x}{x-2} + \frac{x}{x+2} = \frac{2}{x^2-4}\).

Step 2: Identify Common Denominator

The denominators \(x-2\), \(x+2\), and \(x^2-4=(x-2)(x+2)\) suggest that \((x-2)(x+2)\) is the common denominator for the entire equation.

Step 3: Rewrite Each Fraction with the Common Denominator

Rewrite \(\frac{x}{x-2}\) as \(\frac{x(x+2)}{(x-2)(x+2)}\) and \(\frac{x}{x+2}\) as \(\frac{x(x-2)}{(x-2)(x+2)}\). Then the equation becomes:\[\frac{x(x+2) + x(x-2)}{(x-2)(x+2)} = \frac{2}{(x-2)(x+2)}\]

Step 4: Simplify the Numerator

Combine the terms in the numerator: \[x(x+2) + x(x-2) = x^2 + 2x + x^2 - 2x = 2x^2\]Thus, the equation becomes:\[\frac{2x^2}{(x-2)(x+2)} = \frac{2}{(x-2)(x+2)}\]

Step 5: Equate the Numerators

Since the denominators are identical, equate the numerators:\[2x^2 = 2\]

Step 6: Solve for x

Divide both sides of the equation by 2:\[x^2 = 1\]Take the square root of both sides to obtain the solutions for \(x\):\[x = \pm 1\]

Step 7: Verify Solutions

Substitute \(x = 1\) and \(x = -1\) back into the original equation to ensure they are defined and valid. For both values, the equation holds true and both are outside the domain restrictions \(x eq \pm2\).

Key Concepts

Equation Solving

Equation solving is a fundamental aspect of algebra, allowing us to find unknown values. In this exercise, we have an equation involving fractions and complex numbers.
Finding the solution for the original equation involves multiple steps. First, you must rewrite the equation from its original state using proper mathematical notation. This particular equation with complex numbersis expressed as:

• \(x(x-2)^{-1} + x(x+2)^{-1} = 2(x^2-4)^{-1}\)

The goal is to isolate the variable, \(x\), ultimately finding all values that satisfy the equation with zero unknowns.
The equation requires an understanding of algebraic manipulation and comprehension of complex numbers, leading to the final solution: \(x = \pm 1\). Each value of \(x\) will be tested to ensure it fits within any restrictions imposed by denominators.

Fraction Notation

Understanding fraction notation is crucial when dealing with complex numbers equations like the one in this exercise.
Fractions consist of a numerator and a denominator. In the given equation, the fractions are:

• \(\frac{x}{x-2}\)
• \(\frac{x}{x+2}\)
• \(\frac{2}{x^2-4}\)

Each part of a fraction must be treated with care, especially knowing what each term represents.
In equation-solving with fractions, it's often necessary to rewrite those fractions in a different form—by manipulating numerators and denominators—to consolidate or eliminate terms.
This transformation facilitates the equation into a solvable form, making the job of finding common denominatorsmuch more efficient.

Common Denominator

Finding a common denominator is a key step in solving equations involving multiple fractions. In this context, identifying a shared denominator enables you to equate terms across the equation.
Here, the denominators are \(x-2\), \(x+2\), and \(x^2-4\). Observing these denominators immediately suggests the common denominator is \((x-2)(x+2)\).

• This expression results from the product of the two binomials: \((x-2)(x+2) = x^2-4\).

Using this common denominator, every term in the equation can be aligned, ultimately crafting an equivalent equation easier to work with.
Once a common denominator is reached, the next steps involve rewriting and simplifying equations to focus on the numerators.

Numerator Simplification

After achieving a common denominator, attention shifts to the numerators of the fractions.In this exercise, once the fractions are rewritten over the common denominator \((x-2)(x+2)\),the next task is to simplify the numerators before solving for \(x\).
For the given exercise, the rewritten numerator becomes:

• \(x(x+2) + x(x-2)\)

Breaking it down, each term is expanded and simplified:

• \(x(x+2) = x^2 + 2x\)
• \(x(x-2) = x^2 - 2x\)

Combining these results:

• \(x^2 + 2x + x^2 - 2x = 2x^2\)

With a simplified numerator, the equation becomes easier to handle, leading to the final solution where numerators are equated, isolating the variable \(x\).