Problem 26 - Algebra and Trigonometry

Question

1–54 ? Find all real solutions of the equation. $$ \left(\frac{x}{x+2}\right)^{2}=\frac{4 x}{x+2}-4 $$

Step-by-Step Solution

Answer

The real solution is \( x = -4 \).

Step 1: Simplify the Right Side

First, we simplify the right-hand side of the equation. We have: \( \frac{4x}{x+2} - 4 \). We rewrite \(4\) as \( \frac{4(x+2)}{x+2} \). This gives us: \[ \frac{4x}{x+2} - \frac{4(x+2)}{x+2} = \frac{4x - 4(x+2)}{x+2}. \] Simplifying further: \[ \frac{4x - 4x - 8}{x+2} = \frac{-8}{x+2}. \] So the equation becomes \( \left(\frac{x}{x+2}\right)^{2} = \frac{-8}{x+2} \).

Step 2: Cross Multiply

To eliminate the denominators, multiply both sides by \( (x+2)^2 \): \[ \left(\frac{x}{x+2}\right)^{2} \cdot (x+2)^2 = \frac{-8}{x+2} \cdot (x+2)^2. \] This simplifies to: \[ x^2 = -8(x+2). \]

Step 3: Expand and Simplify

Expand the right side of the equation: \[ x^2 = -8x - 16. \] Re-arrange the equation to bring all terms to one side: \[ x^2 + 8x + 16 = 0. \]

Step 4: Factor the Quadratic Equation

Factor the quadratic equation \( x^2 + 8x + 16 = 0 \). Notice that \( (x+4)^2 = x^2 + 8x + 16 \). Therefore, we can write \[ (x+4)^2 = 0. \]

Step 5: Solve for x

Since \( (x+4)^2 = 0 \), take the square root of both sides to solve for \( x \): \[ x+4 = 0. \] Thus, \( x = -4 \).

Step 6: Verify the Solution

Substitute \( x = -4 \) back into the original equation to verify it satisfies \( \left(\frac{x}{x+2}\right)^{2} = \frac{-8}{x+2} \). For \( x = -4 \): \( \left(\frac{-4}{-4+2}\right)^{2} = \frac{-8}{-4+2} \rightarrow \left(\frac{-4}{-2}\right)^{2} = \frac{-8}{-2} \rightarrow 4 = 4. \) Both sides equal, confirming \( x = -4 \) is a solution.

Key Concepts

Cross-Multiplication

Cross-multiplication is a technique that can turn a rational equation, which involves fractions, into an easier-to-solve polynomial equation. In this problem, cross-multiplication is used to simplify the equation by removing the fractions from both sides.This process involves multiplying each side of the equation by the denominator of the other side. Here, in the equation \( \left(\frac{x}{x+2}\right)^{2} = \frac{-8}{x+2} \), you multiply both sides by \((x+2)^2\) to clear the fractions. This results in:

• Left side: \( \left(\frac{x}{x+2}\right)^{2} \cdot (x+2)^2 = x^2 \)
• Right side: \( \frac{-8}{x+2} \cdot (x+2)^2 = -8(x+2) \)

These moves allow you to solve equations without dealing with complex fractions directly, making it easier to work with algebraic expressions cleanly.

Factoring

Factoring is a method used to solve quadratic equations by expressing them as a product of their binomial components. In this exercise, after using cross-multiplication and moving all terms to one side of the equation, we are left with a quadratic equation: \[ x^2 + 8x + 16 = 0. \]Recognizing this quadratic as a perfect square trinomial is the next key step. A perfect square trinomial can be written as the square of a binomial expression. Here:

• The trinomial \( x^2 + 8x + 16 \) can be rewritten as \( (x+4)^2 \).

This means that (x+4) is squared to give the original quadratic equation. Factoring crucially simplifies solving the equation, as you can now set each factor to zero and solve for \( x \). Since in our case, \( (x+4)^2 = 0 \), it follows that \( x+4 = 0 \) or simply \( x = -4 \). This method of solving quadratics works efficiently for expressions that can be easily factored into perfect squares or simple binomials.

Verifying Solutions

Verifying solutions involves substituting the solution back into the original equation to ensure it satisfies the equation. This step is crucial, especially in equations involving variables in denominators, as the solutions must be checked to ensure they do not produce undefined operations, like division by zero.For this problem, once \( x = -4 \) was found as the solution, it's substituted back into the original equation: \[ \left(\frac{x}{x+2}\right)^{2} = \frac{4x}{x+2} - 4. \]Replace \( x \) with \(-4\):

• Left side: \( \left(\frac{-4}{-4+2}\right)^2 = \left(\frac{-4}{-2}\right)^2 = 4 \)
• Right side: \( \frac{-8}{-2} = 4 \)

Both simplified sides equal 4, confirming that the solution \( x = -4 \) holds true in the original equation. This verification step ensures that all solutions are valid and do not cause any mathematical errors, completing the solving process accurately.