Problem 66 - Algebra for College Students

Question

Solve each equation. $$\frac{x+4}{x^{3}+8}+\frac{x+2}{x^{2}-2 x+4}=\frac{11}{2 x+4}$$

Step-by-Step Solution

Answer

The equation has no real solutions.

Step 1: Simplify the Denominators

First, factorize the denominators. Notice that \(x^3 + 8 = (x+2)(x^2 - 2x + 4)\) and \(2x + 4 = 2(x+2)\). The equation becomes: \[\frac{x+4}{(x+2)(x^2-2x+4)} + \frac{x+2}{x^2 - 2x + 4} = \frac{11}{2(x+2)}.\]

Step 2: Find a Common Denominator

Identify the common denominator of all fractions. The least common denominator (LCD) is \2(x+2)(x^2-2x+4)\. Rewrite each fraction with this common denominator: \[\frac{2(x+4)}{2(x+2)(x^2-2x+4)} + \frac{2(x+2)}{2(x+2)(x^2-2x+4)} = \frac{11(x^2-2x+4)}{2(x+2)(x^2-2x+4)}.\]

Step 3: Combine the Numerators

Since the denominators are the same, combine the numerators: \[\frac{2(x+4) + 2(x+2)}{2(x+2)(x^2-2x+4)} = \frac{11(x^2-2x+4)}{2(x+2)(x^2-2x+4)}.\]

Step 4: Simplify the Equation

Combine and simplify the numerators: \[\frac{2x + 8 + 2x + 4}{2(x+2)(x^2-2x+4)} = \frac{11(x^2-2x+4)}{2(x+2)(x^2-2x+4)} \Rightarrow \frac{4x + 12}{2(x+2)(x^2-2x+4)} = \frac{11(x^2-2x+4)}{2(x+2)(x^2-2x+4)}.\]

Step 5: Clear the Denominator

Cancel out the common denominator and solve for x: \[4x + 12 = 11(x^2 - 2x + 4).\] Simplify: \[4x + 12 = 11x^2 - 22x + 44 \Rightarrow 0 = 11x^2 - 26x + 32.\]

Step 6: Solve the Quadratic Equation

Use the quadratic formula \x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\ where \a = 11, b = -26,\ and \c = 32\. Calculate the discriminant: \[b^2 - 4ac = (-26)^2 - 4(11)(32) = 676 - 1408 = -732.\] Since the discriminant is negative, the equation has no real solutions.

Key Concepts

Factoring Polynomials

To begin solving a rational equation like the one in this exercise, it's crucial to factor the polynomials in the denominators since this helps simplify and find a common denominator. For example, the polynomial \(x^3 + 8\) can be factored as \((x+2)(x^2-2x+4)\) by recognizing it as a sum of cubes. The polynomial \(2x + 4\) can be factored as \(2(x+2)\). Simplifying the factors allows for clear identification of the least common denominator (LCD), which significantly eases the process of combining fractions later on.

Common Denominators

After factoring the polynomials, the next step is to identify the least common denominator (LCD). The LCD is crucial because it allows us to rewrite each fraction with a common denominator, making addition or subtraction of the fractions possible. Considering the exercise's denominators, the LCD is \(2(x+2)(x^2-2x+4)\). Rewriting each term using the LCD is crucial:

• Make sure each fraction is expressed with this common denominator.
• This rewriting step helps in combining the fractions smoothly as shown in later steps.

Combining Fractions

Once all fractions share a common denominator, you can combine their numerators. Using the exercise as an example, combine the fractions: \[\frac{2(x+4)}{2(x+2)(x^2-2x+4)} + \frac{2(x+2)}{2(x+2)(x^2-2x+4)} = \frac{11(x^2-2x+4)}{2(x+2)(x^2-2x+4)}.\] Since they have a common denominator, the numerators can be added: \[\frac{2(x+4) + 2(x+2)}{2(x+2)(x^2-2x+4)}. This simplifies to \frac{4x + 12}{2(x+2)(x^2-2x+4)}= \frac{11(x^2-2x+4)}{2(x+2)(x^2-2x+4)}.\] Always simplify the result and ensure both sides of the equation are comparable.

Quadratic Formula

Finally, solving the equation often involves using the quadratic formula, especially if the simplified equation is quadratic. The standard form of the quadratic equation is \(ax^2 + bx + c = 0\). In this exercise, the simplified equation is \(11x^2 - 26x + 32 = 0\). The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is employed here. Calculating the discriminant, \(b^2 - 4ac\): \(-26^2 - 4(11)(32) = 676 - 1408 = -732\). A negative discriminant indicates there are no real solutions for the equation, meaning it has no intersection points with the real number line.