Problem 25 - A Graphical Approach to College Algebra

Question

Find all complex solutions for each equation by hand. Do not use a calculator. $$\frac{1}{x+2}+\frac{3}{x+7}=\frac{5}{x^{2}+9 x+14}$$

Step-by-Step Solution

Answer

There are no valid complex solutions.

Step 1: Identify the denominators

The denominators in the equation are \(x + 2\), \(x + 7\), and \(x^2 + 9x + 14\). Notice that \(x^2 + 9x + 14\) can be factored into \((x+2)(x+7)\).

Step 2: Determine the least common denominator (LCD)

The least common denominator is \((x+2)(x+7)\), since both \(x+2\) and \(x+7\) are present as factors in \(x^2 + 9x + 14\).

Step 3: Clear the fractions by multiplying through by the LCD

Multiply every term by \((x+2)(x+7)\): \[(x+2)(x+7) \cdot \left(\frac{1}{x+2}\right) + (x+2)(x+7) \cdot \left(\frac{3}{x+7}\right) = (x+2)(x+7) \cdot \left(\frac{5}{x^2 + 9x + 14}\right)\]This simplifies to:\[(x+7) + 3(x+2) = 5\]

Step 4: Simplify and solve the resulting equation

Combine like terms:\[x + 7 + 3x + 6 = 5\]Combine the \(x\) terms and the constants:\[4x + 13 = 5\]

Step 5: Solve for \(x\)

Subtract 13 from both sides:\[4x = 5 - 13\]\[4x = -8\]Divide by 4:\[x = -2\]

Step 6: Check the solution in the original equation

Substitute \(x = -2\) back into the original equation:\[\frac{1}{-2+2} + \frac{3}{-2+7} = \frac{5}{(-2)^2 + 9(-2) + 14}\]The left side becomes undefined due to division by zero (\(\frac{1}{0}\)). Hence, \(x = -2\) is not valid due to division by zero in the first term.

Key Concepts

Least Common Denominator

When dealing with fractions in algebraic equations, it's crucial to have a standardized denominator to simplify the problem. This is where the least common denominator (LCD) comes in.
Finding the LCD allows you to combine and simplify fractions by finding a shared base. For instance, in our equation, the denominators were \(x + 2\), \(x + 7\), and \(x^2 + 9x + 14\).

• The first step is to factorize the expression \(x^2 + 9x + 14\) into \((x+2)(x+7)\).
• Now, it's clear that the LCD is \((x+2)(x+7)\), which combines all the distinct factors from the denominators.

This step helps eliminate fractions by allowing each term to be expressed over the same common denominator.

Factoring Quadratics

Factoring quadratics is a vital skill in simplifying polynomial expressions. In our case, the quadratic \(x^2 + 9x + 14\) needed to be factored for further simplification.

• To factorize, look for two numbers that multiply to give the constant term (14) and add to give the middle term (9).
• These numbers are 2 and 7, so the quadratic factors to \((x+2)(x+7)\).

Learning to quickly identify factors can help streamline solving polynomial equations.

Fractions

Fractions in equations often add a layer of complexity, but understanding them is crucial. Fractions consist of a numerator and a denominator.
In algebra, fractions represent division and helping to manage different quantities. To solve equations involving fractions:

• Multiply every term by the least common denominator (LCD) to clear the fractions. This allows you to work with simpler expressions.
• Once the fractions are cleared, solve the remaining equation normally.

In our example equation, after employing the LCD, fractions were removed, simplifying the expression significantly.

Solving Equations

After simplifying an equation by clearing fractions or factoring expressions, the next step is solving it. Solve equations by isolating the variable.
For linear equations, follow these steps:

• Combine like terms if necessary.
• Get all variable terms on one side and constant terms on the other.
• Solve for the variable by performing inverse operations such as addition, subtraction, multiplication, or division.

In the original exercise, after clearing fractions and simplifying, we solved for \(x\) but found that \(x = -2\) does not satisfy the original equation because it results in division by zero.