Question 178500
Let Length=x and Width=x-9  Note: the order does not matter, but the length is usually the longer of the two.



{{{A=LW}}} Start with the area of a rectangle formula



{{{220=x(x-9)}}} Plug in {{{A=220}}}, {{{L=x}}} and {{{W=x-9}}}



{{{220=x^2-9x}}} Distribute



{{{0=x^2-9x-220}}} Subtract 220 from both sides.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-9}}}, and {{{c=-220}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-9) +- sqrt( (-9)^2-4(1)(-220) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-9}}}, and {{{c=-220}}}



{{{x = (9 +- sqrt( (-9)^2-4(1)(-220) ))/(2(1))}}} Negate {{{-9}}} to get {{{9}}}. 



{{{x = (9 +- sqrt( 81-4(1)(-220) ))/(2(1))}}} Square {{{-9}}} to get {{{81}}}. 



{{{x = (9 +- sqrt( 81--880 ))/(2(1))}}} Multiply {{{4(1)(-220)}}} to get {{{-880}}}



{{{x = (9 +- sqrt( 81+880 ))/(2(1))}}} Rewrite {{{sqrt(81--880)}}} as {{{sqrt(81+880)}}}



{{{x = (9 +- sqrt( 961 ))/(2(1))}}} Add {{{81}}} to {{{880}}} to get {{{961}}}



{{{x = (9 +- sqrt( 961 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (9 +- 31)/(2)}}} Take the square root of {{{961}}} to get {{{31}}}. 



{{{x = (9 + 31)/(2)}}} or {{{x = (9 - 31)/(2)}}} Break up the expression. 



{{{x = (40)/(2)}}} or {{{x =  (-22)/(2)}}} Combine like terms. 



{{{x = 20}}} or {{{x = -11}}} Simplify. 



So the possible answers are {{{x = 20}}} or {{{x = -11}}} 

  

However, since you can't have a negative length, this means that only solution is {{{x=20}}}



So the length is {{{L=x=20}}} and the width is {{{W=x-9=20-9=11}}}



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Answer:


So the length is 20 ft and the width is 11 ft