Solve each equation. 2/x+2 + 1/x+4 = 4/x²+6x+8

Verified step by step guidance

1. Identify the given equation: \(\frac{2}{x+2} + \frac{1}{x+4} = \frac{4}{x^{2} + 6x + 8}\).
2. Factor the quadratic expression in the denominator on the right side: \(x^{2} + 6x + 8 = (x+2)(x+4)\).
3. Rewrite the equation using the factored form: \(\frac{2}{x+2} + \frac{1}{x+4} = \frac{4}{(x+2)(x+4)}\).
4. Multiply both sides of the equation by the common denominator \((x+2)(x+4)\) to eliminate the fractions.
5. After clearing denominators, simplify the resulting equation and solve the resulting linear equation for \(x\).

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and/or denominator are polynomials. Understanding how to manipulate these expressions, including simplifying and finding common denominators, is essential for solving equations involving rational terms.

Finding a Common Denominator

To add or subtract rational expressions, you must find a common denominator, often the least common denominator (LCD). This involves factoring denominators and rewriting each fraction with the LCD to combine terms effectively.

Solving Rational Equations

Solving rational equations involves eliminating denominators by multiplying both sides by the LCD, then solving the resulting polynomial equation. It's important to check for extraneous solutions that make any denominator zero.