SOLUTION: Two lines have slopes 2k-4 and k+6. What value(s) of k will produce perpendicular lines? I think the answer is k=-4+or- sqrt66/2????????

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Two lines have slopes 2k-4 and k+6. What value(s) of k will produce perpendicular lines? I think the answer is k=-4+or- sqrt66/2????????      Log On


   

Question 132645: Two lines have slopes 2k-4 and k+6. What value(s) of k will produce perpendicular lines? I think the answer is k=-4+or- sqrt66/2????????
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Their slopes will be perpendicular when their product is -1. So we have the equation

%282k-4%29%2A%28k%2B6%29=-1


2k%5E2%2B8k-24=-1 Foil


2k%5E2%2B8k-23=0 Add 1 to both sides

Let's use the quadratic formula to solve for k:


Starting with the general quadratic

ak%5E2%2Bbk%2Bc=0

the general solution using the quadratic equation is:

k+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29



So lets solve 2%2Ak%5E2%2B8%2Ak-23=0 ( notice a=2, b=8, and c=-23)




k+=+%28-8+%2B-+sqrt%28+%288%29%5E2-4%2A2%2A-23+%29%29%2F%282%2A2%29 Plug in a=2, b=8, and c=-23



k+=+%28-8+%2B-+sqrt%28+64-4%2A2%2A-23+%29%29%2F%282%2A2%29 Square 8 to get 64



k+=+%28-8+%2B-+sqrt%28+64%2B184+%29%29%2F%282%2A2%29 Multiply -4%2A-23%2A2 to get 184



k+=+%28-8+%2B-+sqrt%28+248+%29%29%2F%282%2A2%29 Combine like terms in the radicand (everything under the square root)



k+=+%28-8+%2B-+2%2Asqrt%2862%29%29%2F%282%2A2%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



k+=+%28-8+%2B-+2%2Asqrt%2862%29%29%2F4 Multiply 2 and 2 to get 4

So now the expression breaks down into two parts

k+=+%28-8+%2B+2%2Asqrt%2862%29%29%2F4 or k+=+%28-8+-+2%2Asqrt%2862%29%29%2F4


Now break up the fraction


k=-8%2F4%2B2%2Asqrt%2862%29%2F4 or k=-8%2F4-2%2Asqrt%2862%29%2F4


Simplify


k=-2%2Bsqrt%2862%29%2F2 or k=-2-sqrt%2862%29%2F2


So these expressions approximate to

k=1.93700393700591 or k=-5.93700393700591


So our solutions are:
k=1.93700393700591 or k=-5.93700393700591