Question 202019
You have the correct equation, but made a mistake somewhere in the second step....




{{{1/(1+t)+1/t=1/(6/5)}}} Start with the given equation.



{{{1/(1+t)+1/t=5/6}}} Rewrite {{{1/(6/5)}}} as {{{5/6}}}



{{{6t*cross((1+t))(1/cross((1+t)))+6*cross(t)(1+t)(1/cross(t))=cross(6)t(1+t)(5/cross(6))}}} Multiply EVERY term by the LCD {{{6t(1+t)}}} to clear out the fractions (to make things simpler).



{{{6t+6(1+t)=5t(1+t)}}} Simplify



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



{{{6t+6+6t-5t-5t^2=0}}} Get all terms to the left side


 
{{{-5t^2+7t+6=0}}} Combine like terms.



Notice that the quadratic {{{-5t^2+7t+6}}} is in the form of {{{At^2+Bt+C}}} where {{{A=-5}}}, {{{B=7}}}, and {{{C=6}}}



Let's use the quadratic formula to solve for "t":



{{{t = (-B +- sqrt( B^2-4AC ))/(2A)}}} Start with the quadratic formula



{{{t = (-(7) +- sqrt( (7)^2-4(-5)(6) ))/(2(-5))}}} Plug in  {{{A=-5}}}, {{{B=7}}}, and {{{C=6}}}



{{{t = (-7 +- sqrt( 49-4(-5)(6) ))/(2(-5))}}} Square {{{7}}} to get {{{49}}}. 



{{{t = (-7 +- sqrt( 49--120 ))/(2(-5))}}} Multiply {{{4(-5)(6)}}} to get {{{-120}}}



{{{t = (-7 +- sqrt( 49+120 ))/(2(-5))}}} Rewrite {{{sqrt(49--120)}}} as {{{sqrt(49+120)}}}



{{{t = (-7 +- sqrt( 169 ))/(2(-5))}}} Add {{{49}}} to {{{120}}} to get {{{169}}}



{{{t = (-7 +- sqrt( 169 ))/(-10)}}} Multiply {{{2}}} and {{{-5}}} to get {{{-10}}}. 



{{{t = (-7 +- 13)/(-10)}}} Take the square root of {{{169}}} to get {{{13}}}. 



{{{t = (-7 + 13)/(-10)}}} or {{{t = (-7 - 13)/(-10)}}} Break up the expression. 



{{{t = (6)/(-10)}}} or {{{t =  (-20)/(-10)}}} Combine like terms. 



{{{t = -3/5}}} or {{{t = 2}}} Simplify. 



So the <i>possible</i> solutions are {{{t = -3/5}}} or {{{t = 2}}} 

  


However, a negative time value doesn't make much sense. So we can ignore {{{t = -3/5}}}



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Answer:


So the solution is {{{t = 2}}} which means that it takes Steve 2 hours if he were working alone.