Problem 52 - College Algebra

Question

Add or subtract as indicated. $$ \frac{x}{x^{2}-2 x-24}-\frac{x}{x^{2}-7 x+6} $$

Step-by-Step Solution

Answer

The processed expression is \( \frac{-5x}{(x-6)(x+4)(x-1)} \)

Step 1: Factor the Denominator

The denominators are quadratic expressions which can be factored. The expression \(x^{2}-2 x-24\) can be factored as \( (x-6)(x+4) \), and the expression \(x^{2}-7 x+6\) as \( (x-6)(x-1) \)

Step 2: Find the Common Denominator

To add or subtract fractions, you need a common denominator. The common denominator of the two expressions is the least common multiple of the denominators, which is \( (x-6)(x+4)(x-1) \)

Step 3: Make the Denominators the Same and Subtract

Multiply each fraction by the needed factor to make the denominator the same as the common denominator, and then subtract the numerators:\( \frac{x*(x-1)}{(x-6)(x+4)(x-1)}-\frac{x*(x+4)}{(x-6)(x+4)(x-1)} \)

Step 4: Simplify

Simplify the expression by subtracting the numerators. Distribute the multiple to get:\( \frac{x(x-1)-x(x+4)}{(x-6)(x+4)(x-1)} = \frac{x^2 - x - x^2 - 4x}{(x-6)(x+4)(x-1)} \)

Step 5: Further Simplify the Numerator

Combine similar terms in the numerator to get the final simplified expression: \( \frac{-5x}{(x-6)(x+4)(x-1)} \)

Key Concepts

Factoring Quadratics

Quadratic expressions are polynomials of degree 2, which means the highest power of the variable is 2. To facilitate operations such as addition or subtraction of rational expressions, it is often necessary to factor these quadratics. Factoring means writing the expression as a product of its irreducible factors.

For example, a common method is to express the quadratic equation in the form of:

• \( ax^2 + bx + c \)

In this particular case, the goal is to express it as a product of two binomials such as:

• \( (px + q)(rx + s) \)

Finding such factored forms often involves identifying two numbers that multiply to \( ac \) (the product of the leading coefficient and the constant term) and add to \( b \).

For instance, to factor \( x^2 - 2x - 24 \), we identify the numbers 4 and -6 which multiply to -24 and add up to -2, leading us to \( (x-6)(x+4) \). This way, rational expressions become much easier to handle.

Least Common Denominator

Working with fractions, especially those involving variables, typically involves ensuring the expressions have like denominators. This allows you to perform addition or subtraction. The least common denominator (LCD) is the smallest expression that can be divided evenly by each individual denominator.

In rational expressions, this means factoring each denominator and then combining distinct factors at their highest power. For the expression given, once the quadratics are factored, we identify the LCD by combining all distinct factors:

• \( (x-6) \) is common in both.
• \( (x+4) \) and \( (x-1) \) are unique.

Thus, the LCD for our expression is \( (x-6)(x+4)(x-1) \). Having the denominations aligned is crucial for carrying out further operations.

Subtracting Fractions

Once the denominators are the same, subtracting fractions becomes straightforward. We transform each fraction to have the common denominator by multiplying its numerator and denominator by whatever factor is needed to achieve the LCD.

For our exercise:

• Multiply \( \frac{x}{(x-6)(x+4)} \) by \( \/(x-1) \/(x-1) \).
• Multiply \( \frac{x}{(x-6)(x-1)} \) by \( \/(x+4) \/(x+4) \).

Now you can directly subtract the new numerators because the denominators match. Note how this aligns fractions for direct operation, simplifying our subtraction process. Remember, the new numerator is essentially what's being subtracted.

Simplifying Expressions

After performing subtraction, further simplifying the expression is often necessary. This involves combining like terms and reducing fractions if possible. Simplifying isn't just about making the expression look cleaner; it's about maintaining its equivalency in the most concise form.

In the final step:

• Expand the numerators: \( x(x-1) - x(x+4) \).
• Apply distributive properties: \( x^2 - x - x^2 - 4x \).

Notice that some terms cancel out, leading us to \( -5x \), which is much simpler. This clarity allows us to confidently express results and validate them easily.