Problem 15 - A Graphical Approach to College Algebra

Question

Find all complex solutions for each equation by hand. Do not use a calculator. $$\frac{4}{x^{2}-3 x}-\frac{1}{x^{2}-9}=0$$

Step-by-Step Solution

Answer

The complex solution is \(x = -4\).

Step 1: Find a Common Denominator

To solve the equation \(\frac{4}{x^2 - 3x} - \frac{1}{x^2 - 9} = 0\), we first need to find a common denominator for the fractions. The denominators are \(x^2 - 3x\) and \(x^2 - 9\). Factoring these, we get \(x(x-3)\) and \((x-3)(x+3)\). The least common denominator (LCD) is \(x(x-3)(x+3)\).

Step 2: Rewrite Each Fraction

Express each fraction with this common denominator:\[\frac{4}{x^2 - 3x} = \frac{4(x+3)}{x(x-3)(x+3)}\] and \[\frac{1}{x^2 - 9} = \frac{1x}{x(x-3)(x+3)}\].

Step 3: Combine the Fractions

Now that the fractions have a common denominator, subtract the second fraction from the first:\[\frac{4(x+3) - 1x}{x(x-3)(x+3)} = 0\].This simplifies to:\[\frac{4x + 12 - x}{x(x-3)(x+3)} = 0\],or\[\frac{3x + 12}{x(x-3)(x+3)} = 0\].

Step 4: Set the Numerator to Zero

For the fraction to be zero, its numerator must be zero. Therefore, solve:\[3x + 12 = 0\].Subtract 12 from both sides:\[3x = -12\].Divide by 3:\[x = -4\].

Step 5: Verify and Check for Restrictions

Check the denominator to ensure \(x = -4\) is not a restriction. The restrictions come from \(x(x-3)(x+3)\), which cannot be 0. Thus, \(x eq 0, 3, -3\). Since \(x = -4\) is not one of these restricted values, it is a valid solution.

Key Concepts

Common Denominator

In mathematics, when you encounter an equation involving fractions, finding a common denominator is crucial. This allows you to combine or compare the fractions easily. For the equation \(\frac{4}{x^2 - 3x} - \frac{1}{x^2 - 9} = 0\), the common denominators are the multiplication of the factors found in each original denominator.
\[\text{Steps to find a common denominator:}\]

• Identify the denominators in the fractions. In this case, they are \(x^2 - 3x\) and \(x^2 - 9\).
• Factor these expressions. The first becomes \(x(x-3)\) and the second \((x-3)(x+3)\).
• The common denominator must include each factor the maximum number of times it appears in any of the denominators. Therefore, the common denominator here is \(x(x-3)(x+3)\).

Finding the common denominator helps in expressing the fractions with a unified denominator, paving the way for simpler manipulation in subsequent steps.

Factoring Polynomials

Factoring polynomials is a method used to express a polynomial as the product of its simpler polynomials. It is a key tool for finding common denominators and solving equations, much like what we've done in the exercise.
For example, to factor \(x^2 - 3x\):

• Take out the common factor, which is \(x\), resulting in \(x(x - 3)\).

Similarly, \(x^2 - 9\) is a difference of squares, factoring becomes:

• Recognize it as \((x - 3)(x + 3)\).

Learning these common factoring techniques is vital as it helps simplify expressions, making them easier to solve or manipulate.

Least Common Denominator

The least common denominator (LCD) is essentially the smallest expression that can serve as a common denominator for a set of fractions. To find the LCD, consider each factor in the denominators.

In our exercise, since we factored the denominators to \(x(x-3)\) and \((x-3)(x+3)\), the LCD must include each unique factor at its highest power:

• Both have \((x-3)\).
• You'll add \(x\) from the first and \((x+3)\) from the second.
• The LCD is therefore \(x(x-3)(x+3)\).

The LCD is crucial because it allows you to add or subtract fractions by rewriting each fraction with this shared denominator, thus enabling you to solve the fractional equation easily.

Restrictions on Solutions

While solving equations, especially those with denominators, it's essential to check for restrictions. Restrictions come from values that would make any denominator equal to zero, as division by zero is undefined.
Given our LCD \(x(x-3)(x+3)\), we find restrictions by setting each factor equal to zero:

• \(x = 0\)
• \(x-3 = 0\rightarrow x = 3\)
• \(x+3 = 0\rightarrow x = -3\)

These values are restrictions, meaning the solution cannot be these values. For our solution \(x = -4\), a check confirms that it is not restricted, thereby validating its correctness and ensuring our solution doesn't result in undefined mathematical operations.