Q: Solve for x: 5/(x+4) = 4 + 3/(x-2)
A: I want to clear the denominators on this problem to make it easier to solve. The two denominators are:
(x + 4) and (x – 2)
I am going to multiply the whole equation by (x + 4)(x – 2) like so:
(x + 4)(x – 2) [ 5/(x+4) = 4 + 3/(x-2) ]
Distribute through:
5(x + 4)(x – 2)/(x+4) = 4(x + 4)(x – 2) + 3(x + 4)(x – 2)/(x-2)
Now, cancel things out:
5(x + 4)(x – 2)/(x+4) = 4(x + 4)(x – 2) + 3(x + 4)(x – 2)/(x-2)
5(x – 2) = 4(x + 4)(x – 2) + 3(x + 4)
5x – 10 = 4(x² – 2x + 4x – 8 ) + 3x + 12
5x – 10 = 4x² + 8x – 32 + 3x + 12
5x – 10 = 4x² + 11x – 20
0 = 4x² + 6x – 10
…. Now solve using the quadratic equation…
x = [-b ± √(b² – 4ac)] / (2a)
x = [-6 ± √(6² – 4*4*-10)] / (2*4)
x = [-6 ± √(36 + 160)] / (8)
x = [-6 ± √(196)] / (8)
x = [-6 ± 14] / (8)
x = [-6 + 14] / (8) or x = [-6 – 14] / (8)
x = [8] / (8) or x = [-20] / (8)
x = 1 or x = -5/2