Question 151655
{{{sqrt(t+12)-t=0}}} Start with the given equation.



{{{sqrt(t+12)=t}}} Add {{{t}}} to both sides.



{{{t+12=t^2}}} Square both sides.




{{{-t^2+t+12=0}}} Subtract {{{t^2}}} from both sides.



Notice we have a quadratic equation in the form of {{{at^2+bt+c}}} where {{{a=-1}}}, {{{b=1}}}, and {{{c=12}}}



Let's use the quadratic formula to solve for t



{{{t = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{t = (-(1) +- sqrt( (1)^2-4(-1)(12) ))/(2(-1))}}} Plug in  {{{a=-1}}}, {{{b=1}}}, and {{{c=12}}}



{{{t = (-1 +- sqrt( 1-4(-1)(12) ))/(2(-1))}}} Square {{{1}}} to get {{{1}}}. 



{{{t = (-1 +- sqrt( 1--48 ))/(2(-1))}}} Multiply {{{4(-1)(12)}}} to get {{{-48}}}



{{{t = (-1 +- sqrt( 1+48 ))/(2(-1))}}} Rewrite {{{sqrt(1--48)}}} as {{{sqrt(1+48)}}}



{{{t = (-1 +- sqrt( 49 ))/(2(-1))}}} Add {{{1}}} to {{{48}}} to get {{{49}}}



{{{t = (-1 +- sqrt( 49 ))/(-2)}}} Multiply {{{2}}} and {{{-1}}} to get {{{-2}}}. 



{{{t = (-1 +- 7)/(-2)}}} Take the square root of {{{49}}} to get {{{7}}}. 



{{{t = (-1 + 7)/(-2)}}} or {{{t = (-1 - 7)/(-2)}}} Break up the expression. 



{{{t = (6)/(-2)}}} or {{{t =  (-8)/(-2)}}} Combine like terms. 



{{{t = -3}}} or {{{t = 4}}} Simplify. 



So the possible solutions are {{{t = -3}}} or {{{t = 4}}} 

  
However, we must first check them



Check:


Let's check the possible solution {{{t=-3}}}



{{{sqrt(t+12)-t=0}}} Start with the given equation.



{{{sqrt((-3)+12)-(-3)=0}}} Plug in {{{t=-3}}}.



{{{6=0}}} Evaluate and simplify the left side.



Since the equation is <font size="4"><b>not</b></font> true, this means that {{{t=-3}}} is <font size="4"><b>not</b></font> a solution.


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Let's check the possible solution {{{t=4}}}



{{{sqrt(t+12)-t=0}}} Start with the given equation.



{{{sqrt((4)+12)-(4)=0}}} Plug in {{{t=4}}}.



{{{0=0}}} Evaluate and simplify the left side.



Since the equation is <font size="4"><b>true</b></font>, this means that {{{t=4}}} is a solution.




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Answer:


So the solution is {{{t=4}}}




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{{{5*sqrt(x+1)=3}}} Start with the given equation.



{{{sqrt(x+1)=3/5}}} Divide both sides by 5.



{{{x+1=9/25}}} Square both sides.



{{{x=9/25-1}}} Subtract 1 from both sides.



{{{x=-16/25}}} Combine the fractions.




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Answer:


So the solution is {{{x=-16/25}}}