Problem 16 - Algebra for College Students

Question

Solve each equation. $$ \frac{x}{x-4}-\frac{2}{x+8}=\frac{63}{x^{2}+4 x-32} $$

Step-by-Step Solution

Answer

The solutions are \(x = -11\) and \(x = 5\).

Step 1: Identify and Factor the Denominators

First, recognize that the common denominator on the right side can be factored \( x^2 + 4x - 32 = (x - 4)(x + 8) \). This common denominator is made up of the linear terms found in the first two fractions.

Step 2: Rewrite All Terms with the Common Denominator

Multiply each fraction by appropriate expressions so they all share the common denominator \((x - 4)(x + 8)\). Thus, rewrite: \(\frac{x}{x-4} = \frac{x(x+8)}{(x-4)(x+8)}\) and \(\frac{2}{x+8} = \frac{2(x-4)}{(x - 4)(x + 8)}\).

Step 3: Simplify the Equation

The equation becomes: \( \frac{x(x+8) - 2(x-4)}{(x-4)(x+8)} = \frac{63}{(x-4)(x+8)} \). Now, simplify the numerator: \(x^2 + 8x - 2x + 8 = x^2 + 6x + 8.\)

Step 4: Clear the Fractions by Equating the Numerators

Since the denominators are identical, equate the numerators: \(x^2 + 6x + 8 = 63\).

Step 5: Solve the Quadratic Equation

Rearrange to form a standard quadratic equation: \(x^2 + 6x + 8 - 63 = 0\) which becomes \(x^2 + 6x - 55 = 0\). Factor this to find \((x+11)(x-5) = 0\).

Step 6: Find the Values of x

Solve each factor equal to zero: \(x+11=0\) gives \(x=-11\), and \(x-5=0\) gives \(x=5\).

Step 7: Check for Valid Solutions

The initial constraints: \(x eq 4\) and \(x eq -8\) due to division by zero in the original expression, neither \(-11\) nor \(5\) satisfy these conditions. Meanwhile ensure both solutions don't simplify the denominator alone without multiplication. Thus both solutions are valid.

Key Concepts

Factoring Quadratic Expressions

When dealing with quadratic expressions, it's often helpful to factor them into simpler pieces. Remember that factoring means expressing a quadratic in the form

• \( ax^2 + bx + c = (px + q)(rx + s) \).

Finding the factors allows us to understand the components that multiply together to produce the original expression. In our exercise, we need to factor

• \( x^2 + 4x - 32 \). The expression can be broken into \((x - 4)(x + 8) \).

This means the quadratic can be written as a product of two binomials, which helps simplify operations such as finding common denominators.

Common Denominators in Algebra

In the realm of rational equations, common denominators are essential for simplifying expressions. They allow different fractions to be combined or compared. To find a common denominator, look for the least common multiple of the separate denominators. This might mean factoring each denominator first, as we did in the exercise where we factored

• \( (x - 4) \) and \( (x + 8) \).

Combining together gives a common denominator of

• \((x - 4)(x + 8) \).

Once all fractions share this common base, you can easily manipulate them, such as adding or subtracting, by focusing on the numerators instead.

Solving Quadratic Equations

Solving quadratic equations—the ones that take the form

• \( ax^2 + bx + c = 0 \)—is a fundamental skill in algebra.

One way to solve them is by factoring, as shown in the exercise when we derived

• \((x + 11)(x - 5) = 0 \).

Once factored, set each expression equal to zero:

• \( x + 11 = 0 \) or \( x - 5 = 0 \).

Solving these simple equations gives the roots or solutions of the quadratic, which in this case is

• \( x = -11 \) and \( x = 5 \).

These roots are the values where the quadratic expression equals zero.

Steps for Solving Algebraic Equations

Solving algebraic equations involves a systematic approach:

• 1. Identify and Simplify: Look for opportunities to factor or simplify the terms involved, such as finding a common denominator for rational expressions.
• 2. Rewrite with Common Denominators: This step ensures that all terms can be easily compared or combined. As in our example, multiplying by factors gives all sides the new denominator.
• 3. Simplify and Clear Fractions: Once the denominators are the same, you can focus solely on the numerators. Use simplification steps to handle the equation as a standard algebraic problem.
• 4. Solve Using Factoring: Look to factor the resulting polynomial, then solve by setting each factor to zero as seen with \( (x+11)(x-5) = 0 \).
• 5. Validate Solutions: Always ensure your solutions don't contradict any initial constraints, particularly concerning division by zero or other undefined operations.

Following these steps will guide you through solving complex algebraic equations with confidence.