How do you subtract #\frac{2x}{x^2+10x+25}-\frac{3x}{2x^2+7x-15}#?

Question

Hazel Anglin · Accepted Answer

First, we're going to simplify the expressions a bit. We can do this by factoring both the denominators:
#x^2+10x+25 = (x+5)^2# #2x^2+7x-15=(x+5)(2x-3)# To substract two fractions with a different denominator, we will always have to find the LCM of the denominators. In this case it is #(x+5)^2(2x-3)# since this can evenly be divided by both the denominators. Let's combine what we've got already:
#(2x)/(x+5)^2-(3x)/((x+5)(2x-3))# Now for changing to the LCM:
#(2x-3)/(2x-3)*(2x)/(x+5)^2 - (x+5)/(x+5)(3x)/((x+5)(2x-3))# #= (2x*(2x-3)-3x*(x+5))/((x+5)^2(2x-3))# Distribute the terms: #(4x^2-6x-3x^2-15x)/((x+5)^2(2x-3))# #= (x^2-21x)/((x+5)^2(2x-3))# You could've also done this without factoring, but it would've been way harder. Factoring always makes your life easier! I really hope this helped.

Eva Abramson · Answer

undefined To subtract the given expressions, we need to find a common denominator. The denominators are (x^2+10x+25) and (2x^2+7x-15). The common denominator is the product of these two denominators, which is (x^2+10x+25)(2x^2+7x-15).

Next, we multiply each fraction by the appropriate factor to obtain the common denominator. For the first fraction, we multiply the numerator and denominator by (2x^2+7x-15). For the second fraction, we multiply the numerator and denominator by (x^2+10x+25).

After multiplying, we can combine the numerators over the common denominator. Simplify the resulting expression if necessary.