Problem 53 - College Algebra

Question

Add or subtract as indicated. $$\frac{3 x}{x^{2}+3 x-10}-\frac{2 x}{x^{2}+x-6}$$

Step-by-Step Solution

Answer

\[\frac{x^2-x-10}{(x-2)(x+5)(x+3)}\] is the simplified result of the subtraction.

Step 1: Factor the denominators

Factoring each denominator \(x^2 + 3x - 10\) and \(x^2 + x - 6\), we get \((x-2)(x+5)\) and \((x-2)(x+3)\) respectively. To factor a trinomial \(mx^2+nx+p\), we look for two numbers that multiply to give the product \(mxp\) and add to give \(n\). Here, -2 and 5 add to give 3 and multiply to give -10 for the first denominator, and -2 and 3 multiply to give -6 and add to give 1 for the second denominator.

Step 2: Determine the common denominator

To find the common denominator between \((x-2)(x+5)\) and \((x-2)(x+3)\), we take the union of all the unique factors in the two expressions. The common denominator thus becomes \((x-2)(x+5)(x+3)\).

Step 3: Rewrite the fractions with common denominator

Now the fractions can be rewritten with this common denominator. For the first fraction, multiply both numerator and denominator by \((x+3)\), resulting in \((3x \cdot (x+3)) / ((x-2)(x+5)(x+3))\). For the second, multiply both numerator and denominator by \((x+5)\), resulting in \((2x \cdot (x+5)) / ((x-2)(x+5)(x+3))\).

Step 4: Perform the operation

We then proceed to subtract the two fractions since they now have the same denominator, resulting in \(((3x \cdot (x+3))-(2x \cdot (x+5))) / ((x-2)(x+5)(x+3))\). This simplifies to \(x^2-x-10 / ((x-2)(x+5)(x+3))\).