Solve each equation in Exercises 83–108 by the method of your choice. 3/(x - 3) + 5/(x - 4) = (x² - 20)/(x² - 7x + 12)

Verified step by step guidance

1. Step 1: Analyze the equation and identify the denominators. The denominators are (x - 3), (x - 4), and (x^2 - 7x + 12). Factorize the quadratic denominator x^2 - 7x + 12 into (x - 3)(x - 4). This simplifies the equation to: 3/(x - 3) + 5/(x - 4) = (x^2 - 20)/((x - 3)(x - 4)).
2. Step 2: Determine the least common denominator (LCD) for all terms. The LCD is (x - 3)(x - 4). Rewrite each term in the equation with this common denominator.
3. Step 3: Rewrite the left-hand side of the equation. Multiply the numerator and denominator of 3/(x - 3) by (x - 4), and multiply the numerator and denominator of 5/(x - 4) by (x - 3). This gives: (3(x - 4) + 5(x - 3))/((x - 3)(x - 4)).
4. Step 4: Combine the left-hand side into a single fraction. Expand the numerators: 3(x - 4) becomes 3x - 12, and 5(x - 3) becomes 5x - 15. Add these together to get (3x - 12 + 5x - 15)/((x - 3)(x - 4)) = (8x - 27)/((x - 3)(x - 4)).
5. Step 5: Set the left-hand side equal to the right-hand side. Now the equation is: (8x - 27)/((x - 3)(x - 4)) = (x^2 - 20)/((x - 3)(x - 4)). Since the denominators are the same, equate the numerators: 8x - 27 = x^2 - 20. Rearrange this into a standard quadratic equation: x^2 - 8x + 7 = 0. Solve this quadratic equation using factoring, completing the square, or the quadratic formula.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding how to manipulate these expressions, including finding common denominators and simplifying, is crucial for solving equations involving them. In this problem, the presence of rational expressions requires careful handling to combine and solve the equation effectively.

Finding Common Denominators

To solve equations involving rational expressions, it is often necessary to find a common denominator. This process allows for the combination of fractions into a single expression, making it easier to isolate variables. In the given equation, identifying the least common denominator will facilitate the simplification and solution of the equation.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. This concept is essential in solving equations, particularly when simplifying expressions or finding roots. In this problem, factoring the quadratic expression in the numerator and the denominator will help in simplifying the equation and finding the values of x.