Problem 58 - College Algebra Essentials

Question

Add or subtract as indicated. $$\frac{6 x^{2}+17 x-40}{x^{2}+x-20}+\frac{3}{x-4}-\frac{5 x}{x+5}$$

Step-by-Step Solution

Answer

\(\frac{6x^2-59x+90}{x^2+x-20}\)

Step 1: Simplify the Expressions

Begin by simplifying the expressions. The quadratic expressions in the first term can be factored. Expression \(\frac{6 x^{2}+17 x-40}{x^{2}+x-20}\) can be factored out to \(\frac{6(x+5)(x-2)}{(x-4)(x+5)}\). Now the expression is \(\frac{6(x+5)(x-2)}{(x-4)(x+5)}+\frac{3}{x-4}-\frac{5 x}{x+5}\)

Step 2: Cancel out the common factor

There is a factor of \(x+5\) in the numerator and denominator of the first term, they can be cancelled out. So, the expression becomes \(\frac{6(x-2)}{x-4}+\frac{3}{x-4}-\frac{5 x}{x+5}\)

Step 3: Find a Common Denominator

Next, find a common denominator among all three terms which is \((x-4)(x+5)\). Multiply each term by whatever it lacks to have this common denominator.

Step 4: Simplify the terms

When each term has this common denominator, add or subtract them as indicated: \(frac{6(x-2)(x+5)+(3(x+5)-5x(x-4)}{(x-4)(x+5)}\)

Step 5: Apply Distribution

Apply distribution in the numerator and sum or subtract the terms in the numerator then simplify: \(final result = \frac{6x^2-59x+90}{x^2+x-20}\)