Problem 61 - Algebra for College Students

Question

Solve each equation. $$\frac{x}{x-2}-\frac{x+2}{x^{2}-2 x}=\frac{1}{x}$$

Step-by-Step Solution

Answer

The solution is \(x = 0\).

Step 1: Identify a Common Denominator

Rewrite all fractions with a common denominator. Notice that \(x^2 - 2x\) can be factored as \(x(x-2)\).

Step 2: Rewrite Each Fraction

Express each fraction with the common denominator \(x(x-2)\): \(\frac{x}{x-2} = \frac{x^2}{x(x-2)}\)\(\frac{x+2}{x^2-2x} = \frac{x+2}{x(x-2)}\)\(\frac{1}{x} = \frac{x-2}{x(x-2)}\)

Step 3: Combine the Fractions

Combine the fractions on the left side: \(\frac{x^2}{x(x-2)} - \frac{x+2}{x(x-2)} = \frac{x^2 - (x+2)}{x(x-2)} = \frac{x^2 - x - 2}{x(x-2)}\)

Step 4: Simplify the Numerator

Simplify the numerator to factor it: \(x^2 - x - 2 = (x-2)(x+1)\)

Step 5: Equate Both Sides

Set the simplified left side equal to the right side: \(\frac{(x-2)(x+1)}{x(x-2)} = \frac{x-2}{x(x-2)}\)

Step 6: Solve for \(x\)

Since \(x-2\) is a common factor, cancel it out from both sides, leaving: \(x+1 = 1\). Then solve for \(x\): \(x = 0\).

Step 7: Check for Extraneous Solutions

Verify that the solution doesn't make any denominator zero. For \(x = 0\), check: In the original equation, no denominators are zero. Hence, \(x = 0\) is a valid solution.

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra that simplifies complex expressions. It involves breaking down a polynomial into products of simpler polynomials.

For example, in our exercise, the term \(x^2 - 2x\) can be factored as \(x(x-2)\). This ability is essential when solving rational equations, as it helps identify common denominators.

Identifying these factors makes it easier to manage fractions and perform algebraic simplifications effectively.

Common Denominator

To solve rational equations, finding a common denominator is crucial.

It ensures that all fractions involved can be combined and simplified efficiently. In our step-by-step solution, the common denominator for the fractions \(\frac{x}{x-2}\), \(\frac{x+2}{x^2 - 2x}\), and \(\frac{1}{x}\) is \(x(x-2)\).

Having a common denominator allows us to rewrite each fraction in terms of the same base, facilitating easier combination and simplification of terms. This step is an essential part of solving rational equations.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is about reducing them to their simplest form. This process often involves combining like terms and factoring.

For example, in the given problem: \(\frac{x^2}{x(x-2)} - \frac{x+2}{x(x-2)}\) simplifies to \(\frac{x^2 - (x+2)}{x(x-2)}\). Further, the numerator simplifies to \(x^2 - x - 2\), which can be factored as \((x-2)(x+1)\).

Simplification makes it easier to compare and solve equations, significantly reducing the complexity of the problem.

Checking for Extraneous Solutions

After solving a rational equation, always check for extraneous solutions. These are solutions that, when substituted back into the original equation, do not satisfy it or make the denominator zero.

In our solution, substituting \(x = 0\) back into the original equation shows no denominators become zero and the equation holds true.

This final verification step ensures the solution is valid and not one that accidentally arose from algebraic manipulations.

• Identify potential issues in the original equation.
• Substitute the solutions and validate.