Question 148820
Let x=length of one side and y=length of other side


Since "One side of a rectangular stage is 2 meters longer than the other", this means that {{{y=x+2}}} (note: the order does not matter). Also, since the "diagonal is 10 meters", we can use Pythagoreans Theorem to get


{{{x^2+y^2=10^2}}} which becomes {{{x^2+y^2=100}}}



{{{x^2+y^2=100}}} Start with the second equation.



{{{x^2+(x+2)^2=100}}} Plug in {{{y=x+2}}}



{{{x^2+x^2+4x+4=100}}} Foil.



{{{x^2+x^2+4x+4-100=0}}} Get all terms to the left side.



{{{2x^2+4x-96=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=2}}}, {{{b=4}}}, and {{{c=-96}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(4) +- sqrt( (4)^2-4(2)(-96) ))/(2(2))}}} Plug in  {{{a=2}}}, {{{b=4}}}, and {{{c=-96}}}



{{{x = (-4 +- sqrt( 16-4(2)(-96) ))/(2(2))}}} Square {{{4}}} to get {{{16}}}. 



{{{x = (-4 +- sqrt( 16--768 ))/(2(2))}}} Multiply {{{4(2)(-96)}}} to get {{{-768}}}



{{{x = (-4 +- sqrt( 16+768 ))/(2(2))}}} Rewrite {{{sqrt(16--768)}}} as {{{sqrt(16+768)}}}



{{{x = (-4 +- sqrt( 784 ))/(2(2))}}} Add {{{16}}} to {{{768}}} to get {{{784}}}



{{{x = (-4 +- sqrt( 784 ))/(4)}}} Multiply {{{2}}} and {{{2}}} to get {{{4}}}. 



{{{x = (-4 +- 28)/(4)}}} Take the square root of {{{784}}} to get {{{28}}}. 



{{{x = (-4 + 28)/(4)}}} or {{{x = (-4 - 28)/(4)}}} Break up the expression. 



{{{x = (24)/(4)}}} or {{{x =  (-32)/(4)}}} Combine like terms. 



{{{x = 6}}} or {{{x = -8}}} Simplify. 



So the possible answers are {{{x = 6}}} or {{{x = -8}}} 

  
  
However, since a negative length is not possible, this means that the only answer is {{{x = 6}}}


{{{y=x+2}}} Go back to the first equation.



{{{y=6+2}}} Plug in {{{x = 6}}}.



{{{y=8}}} Add.



So the dimensions of the rectangle are 6 and 8