Question 88652
{{{5/2t-t=3+3/2t}}}


{{{2t(5/2t-t)=2t(3+3/2t)}}} Multiply both sides by 2t


{{{5-2t^2=6t+3}}} Distribute


{{{5-2t^2-6t-3=0}}} Get everything to one side


{{{-2t^2-6t+2=0}}} Combine like terms



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



Starting with the general quadratic


{{{at^2+bt+c=0}}}


the general solution using the quadratic equation is:


{{{t = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}


So lets solve {{{-2*t^2-6*t+2=0}}} ( notice {{{a=-2}}}, {{{b=-6}}}, and {{{c=2}}})


{{{t = (--6 +- sqrt( (-6)^2-4*-2*2 ))/(2*-2)}}} Plug in a=-2, b=-6, and c=2




{{{t = (6 +- sqrt( (-6)^2-4*-2*2 ))/(2*-2)}}} Negate -6 to get 6




{{{t = (6 +- sqrt( 36-4*-2*2 ))/(2*-2)}}} Square -6 to get 36




{{{t = (6 +- sqrt( 36+16 ))/(2*-2)}}} Multiply {{{-4*2*-2}}} to get {{{16}}}




{{{t = (6 +- sqrt( 52 ))/(2*-2)}}} Combine like terms in the radicand (everything under the square root)




{{{t = (6 +- 2*sqrt(13))/(2*-2)}}} Simplify the square root




{{{t = (6 +- 2*sqrt(13))/-4}}} Multiply 2 and -2 to get -4


So now the expression breaks down into two parts


{{{t = (6 + 2*sqrt(13))/-4}}} or {{{t = (6 - 2*sqrt(13))/-4}}}



which split up to



{{{t=+6/-4+2*sqrt(13)/-4}}} or {{{t=+6/-4+2*sqrt(13)/-4}}}



and simplify to



{{{t=-3 / 2-sqrt(13)/ 2}}} or {{{t=-3 / 2+ sqrt(13)/ 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*x^2-6*x+2}}} (just replace t with x), we get:


{{{ graph( 500, 500, -13.302775637732, 10.302775637732, -13.302775637732, 10.302775637732,-2*x^2+-6*x+2) }}}


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