SOLUTION: 5/2t-t=3+3/2t

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Question 88652: 5/2t-t=3+3/2t
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
5%2F2t-t=3%2B3%2F2t

2t%285%2F2t-t%29=2t%283%2B3%2F2t%29 Multiply both sides by 2t

5-2t%5E2=6t%2B3 Distribute

5-2t%5E2-6t-3=0 Get everything to one side

-2t%5E2-6t%2B2=0 Combine like terms


Now let's use the quadratic formula to solve for t:


Starting with the general quadratic

at%5E2%2Bbt%2Bc=0

the general solution using the quadratic equation is:

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

So lets solve -2%2At%5E2-6%2At%2B2=0 ( notice a=-2, b=-6, and c=2)

t+=+%28--6+%2B-+sqrt%28+%28-6%29%5E2-4%2A-2%2A2+%29%29%2F%282%2A-2%29 Plug in a=-2, b=-6, and c=2



t+=+%286+%2B-+sqrt%28+%28-6%29%5E2-4%2A-2%2A2+%29%29%2F%282%2A-2%29 Negate -6 to get 6



t+=+%286+%2B-+sqrt%28+36-4%2A-2%2A2+%29%29%2F%282%2A-2%29 Square -6 to get 36



t+=+%286+%2B-+sqrt%28+36%2B16+%29%29%2F%282%2A-2%29 Multiply -4%2A2%2A-2 to get 16



t+=+%286+%2B-+sqrt%28+52+%29%29%2F%282%2A-2%29 Combine like terms in the radicand (everything under the square root)



t+=+%286+%2B-+2%2Asqrt%2813%29%29%2F%282%2A-2%29 Simplify the square root



t+=+%286+%2B-+2%2Asqrt%2813%29%29%2F-4 Multiply 2 and -2 to get -4

So now the expression breaks down into two parts

t+=+%286+%2B+2%2Asqrt%2813%29%29%2F-4 or t+=+%286+-+2%2Asqrt%2813%29%29%2F-4


which split up to


t=%2B6%2F-4%2B2%2Asqrt%2813%29%2F-4 or t=%2B6%2F-4%2B2%2Asqrt%2813%29%2F-4


and simplify to


t=-3+%2F+2-sqrt%2813%29%2F+2 or t=-3+%2F+2%2B+sqrt%2813%29%2F+2


Which approximate to

t=-3.30277563773199 or t=0.302775637731995


So our solutions are:
t=-3.30277563773199 or t=0.302775637731995

Notice when we graph -2%2Ax%5E2-6%2Ax%2B2 (just replace t with x), we get:



when we use the root finder feature on a calculator, we find that x=-3.30277563773199 and x=0.302775637731995.So this verifies our answer