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Chapter 0: Problem 58

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

Short Answer

Expert verified
The simplified expression is: \(\frac{x^2 + 40x - 25}{(x - 4)(x + 5)}\).

Step by step solution

01

Identify the Common Denominator

Focus on the denominators: \(x^{2} + x - 20\), \(x - 4\), and \(x + 5\). Note that \(x^{2} + x - 20\) can be factorised into \((x-4)(x+5)\) which includes the other two denominators, so the common denominator will be: \(x^{2} + x - 20 = (x-4)(x+5)\).
02

Reform Fraction with the Common Denominator

For the 2nd term, multiply the numerator and the denominator by \((x+5)\) and for the 3rd term multiply the numerator and the denominator by \((x-4)\). You will then have: \(\frac{6x^2 + 17x - 40}{(x-4)(x+5)} + \frac{3(x+5)}{(x - 4)(x + 5)} - \frac{5x(x - 4)}{(x - 4)(x + 5)}\).
03

Simplify the Numerator

Then combine like terms within the numerators. The expression becomes: \(\frac{6x^2 + 17x - 40 + 3x + 15 - 5x^2 + 20x}{(x - 4)(x + 5)}\). Further, we simplify the numerator by combining like terms which gives: \(\frac{x^2 + 40x - 25}{(x - 4)(x + 5)}\).
04

Further Simplification

The whole fraction cannot be simplified anymore. Thus, the result is stayed as this: \(\frac{x^2 + 40x - 25}{(x - 4)(x + 5)}\)

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