How do you combine #(4-3x)/ (16-x^2) + 3/( x-4)#?

Question

Eli Assouline · Accepted Answer

You need to find the lowest common denominator among these two functions. We can try to factor the first function's denominator in order to ease things for us, as shown: #16-x^2=0#
#16=x^2#
#sqrt(16)=x#
#x=+-4#, which means #x-4=0# and #x+4=0#. However, the factors #(x-4)(x+4)# when multiplied, give us the opposite of the first function's denominator: #(x-4)(x+4)=x^2-16# We just need to multiply it  by #(-1)# to validate our equality. So, we can rewrite the whole sum as #(4-3x)/((-1)(x+4)(x-4))+3/(x-4)# Now, our lowest common denominator is #(x+4)(x-4)#. Adopting it as l.c.d., then we have: #((4-3x)+3(-x-4))/((-1)(x+4)(x-4))#=#(4-6x-8)/((-1)(x+4)(x-4))# Distributing our factors in the denominator, we have the shortest answer as follows: #color(green)((-6x-8)/(16-x^2))#

Abigail Allen · Answer

undefined To combine the expressions (4-3x)/(16-x^2) and 3/(x-4), we need to find a common denominator. The common denominator for these expressions is (16-x^2)(x-4).

Next, we multiply the numerator and denominator of the first fraction, (4-3x), by (x-4), and multiply the numerator and denominator of the second fraction, 3, by (16-x^2).

After simplifying the expressions, we can combine the numerators over the common denominator.

The combined expression is [(4-3x)(x-4) + 3(16-x^2)] / (16-x^2)(x-4).