Problem 81 - Intermediate Algebra

Question

Solve equation. If a solution is extraneous, so indicate. \(\frac{x}{x+2}=1-\frac{3 x+2}{x^{2}+4 x+4}\)

Step-by-Step Solution

Answer

The solution is any real number except \(x = -2\), which is extraneous.

Step 1: Identify Expressions

The given equation is \( \frac{x}{x+2} = 1 - \frac{3x+2}{x^2 + 4x + 4} \). We notice that the denominator on the right side \(x^2 + 4x + 4\) can be rewritten, often indicating a common factor for simplification.

Step 2: Factor the Quadratic

Recognize that the quadratic expression \(x^2 + 4x + 4\) can be rewritten as \((x+2)^2\). This will allow simplification of fractions later.

Step 3: Rewrite the Equation

Substitute the factored expression into the equation: \( \frac{x}{x+2} = 1 - \frac{3x+2}{(x+2)^2} \). This presents an opportunity to find a common denominator for the fractions.

Step 4: Combine Fractions on the Right Side

Rewrite the right side with a common denominator: \(1 = \frac{(x+2)^2}{(x+2)^2}\). Therefore, the equation becomes \( \frac{x}{x+2} = \frac{(x+2)^2-(3x+2)}{(x+2)^2} \).

Step 5: Simplify the Numerator

Subtract the numerators: \((x+2)^2 - (3x+2) = x^2 + 4x + 4 - 3x - 2 = x^2 + x + 2\). So the equation becomes \( \frac{x}{x+2} = \frac{x^2 + x + 2}{(x+2)^2} \).

Step 6: Cross-Multiply to Solve

Cross-multiplying gives: \(x(x+2)^2 = (x+2)(x^2 + x + 2) \). Simplify: \(x(x^2+4x+4) = x^3+3x^2+4x+2x+4 \). Therefore, both sides are equal, so the equation simplifies to an identity.

Step 7: Analyze Solutions and Extraneous Roots

The simplification shows an identity, suggesting \(x\) can take any value, except denominators become zero. Thus \(x = -2\) is extraneous as a solution, because it makes the original denominators zero.

Key Concepts

Understanding Quadratic Factorization

Quadratic factorization is a crucial concept when solving rational equations, particularly when dealing with complex expressions. In the example equation \(\frac{x}{x+2} = 1 - \frac{3x+2}{x^{2}+4x+4}\), the quadratic expression, \(x^2 + 4x + 4\), in the denominator can be simplified by factorization. This expression is a perfect square trinomial, which can be rewritten as \((x+2)^2\).

• This rewriting aids in simplifying complex fractions.
• Recognizing this factorization helps to identify common denominators quickly.
• Efficient factorization simplifies the process of solving the equation.

By converting the quadratic expression into its factored form, we prepare the equation for further simplification, ultimately leading to an easier solution process.

The Mechanics of Cross-Multiplication

Cross-multiplication is a powerful technique for simplifying equations involving fractions. It's an efficient way to eliminate the fractional terms, making it easier to solve for the unknown variable. Once we simplified our equation to \(\frac{x}{x+2} = \frac{x^2 + x + 2}{(x+2)^2}\), we employed cross-multiplication.

• This technique follows the principle of multiplying both sides of the equation, ensuring the fractional terms are effectively removed.
• We cross-multiply by multiplying the numerator of one side by the denominator of the other and vice versa.
• This provides a path toward finding a simpler equation quickly without fractions, resulting in the equation \(x(x+2)^2 = (x+2)(x^2 + x + 2)\).

Through cross-multiplication, the equation boils down to simpler terms, making it more manageable and leading us closer to the solution.

Recognizing and Dealing with Extraneous Solutions

While solving rational equations, it is essential to identify and recognize any potential extraneous solutions. These are solutions that may emerge during the algebraic process but do not satisfy the original equation. In this exercise, we reach an identity, indicating that all values of \(x\) should theoretically solve the equation.

• One must check the conditions under which the original denominators become zero.
• In this problem, \(x = -2\) would make both original denominators undefined, making it an extraneous solution.
• Extraneous solutions typically arise from the act of cross-multiplying or squaring both sides of an equation during the solving process.

After solving the equation, it's critical to re-evaluate each possible solution against the original equation to confirm whether they are valid or extraneous. This prevents inaccuracies in your final answer.