How do you solve # 1/ (x-2) + (2x)/ ((x-2)(x-8)) = x/ (2(x-8)) #?

1 Answer
Apr 22, 2017

Answer:

Multiply both sides by the least common multiple of the numerators.
Solve the resulting quadratic.
Check.

Explanation:

Given: #1/ (x-2) + (2x)/ ((x-2)(x-8)) = x/ (2(x-8));x!=2,x!=8#

Multiply both sides by #2(x-2)(x-8)#

#2(x-8) + 2(2x) = x(x-2);x!=2,x!=8#

#2x-16 + 4x = x^2-2x;x!=2,x!=8#

Combine like terms:

#0 = x^2-8x+16;x!=2,x!=8#

Neither 2 nor 8 are roots, therefore, we drop the restrictions:

#0 = x^2-8x+16#

The above factors into a perfect square:

#(x - 4)^2 = 0#

#x = 4#

check:

#1/ (4-2) + (2(4))/ ((4-2)(4-8)) = 4/ (2(4-8))#
#1/2 + 8/ ((2)(-4)) = 2/(-4)#
#-1/2 = -1/2#

This checks.