How do you subtract #9/(x^2-9) - 5/ (x+3)#?

Question

How do you subtract #9/(x^2-9) -  5/ (x+3)#?

Ethan Adkins · Accepted Answer

I have shown every step in 'over the top' detail so you can see where everything comes from.

#- (5x-24)/(x^2-9)# When presented with questions you look for links which means you have to build up a 'toolbox' of remembered facts and techniques: Given:# 9/(x^2-9) - 5/(x+3)# .............(1) Consider #x^2-9# ....................(2) But #9 = 3^2# .....................(3) Substitute (3) into (2) giving: #x^2 - 3^3# But #(x^2-3^2)=(x-3)(x+3)#........(4) Substitute (4) into (1) giving: #9/((x-3)(x+3)) - 5/(x+3)# Now both denominators have something in common so we can combine them #(9-5(x-3))/((x+3)(x-3))# #(9-5x+15)/((x+3)(x-3))# #((-5x)+24)/ ((x+3)(x-3))# .................(5) but #(-5x+24) = -(5x-24)#...............(6) substituting (6) into (5) and rewriting the expression gives: #- (5x-24)/(x^2-9)# If you found this helpful please click on the thumbs up

Isaiah Anthony · Answer

undefined To subtract the given expressions, we need to find a common denominator. The common denominator for the two expressions is (x+3)(x-3).

Next, we multiply the numerator and denominator of the first fraction, 9/(x^2-9), by (x+3) to get 9(x+3).

Similarly, we multiply the numerator and denominator of the second fraction, 5/(x+3), by (x-3) to get 5(x-3).

Now, we can rewrite the subtraction as a single fraction: (9(x+3) - 5(x-3))/(x+3)(x-3).

Expanding the numerator, we have (9x + 27 - 5x + 15)/(x+3)(x-3).

Combining like terms, we get (4x + 42)/(x+3)(x-3).

Therefore, the simplified expression is 4x + 42 divided by (x+3)(x-3).