Question 338993
"A rectangular garden is 10ft longer than it is wide. Its area is 6375 ft^2" means that {{{x(x+10)=6375}}}




{{{x(x+10)=6375}}} Start with the given equation.



{{{x^2+10x=6375}}} Distribute.



{{{x^2+10x-6375=0}}} Subtract 6375 from both sides.



Notice that the quadratic {{{x^2+10x-6375}}} is in the form of {{{Ax^2+Bx+C}}} where {{{A=1}}}, {{{B=10}}}, and {{{C=-6375}}}



Let's use the quadratic formula to solve for "x":



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



{{{x = (-(10) +- sqrt( (10)^2-4(1)(-6375) ))/(2(1))}}} Plug in  {{{A=1}}}, {{{B=10}}}, and {{{C=-6375}}}



{{{x = (-10 +- sqrt( 100-4(1)(-6375) ))/(2(1))}}} Square {{{10}}} to get {{{100}}}. 



{{{x = (-10 +- sqrt( 100--25500 ))/(2(1))}}} Multiply {{{4(1)(-6375)}}} to get {{{-25500}}}



{{{x = (-10 +- sqrt( 100+25500 ))/(2(1))}}} Rewrite {{{sqrt(100--25500)}}} as {{{sqrt(100+25500)}}}



{{{x = (-10 +- sqrt( 25600 ))/(2(1))}}} Add {{{100}}} to {{{25500}}} to get {{{25600}}}



{{{x = (-10 +- sqrt( 25600 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (-10 +- 160)/(2)}}} Take the square root of {{{25600}}} to get {{{160}}}. 



{{{x = (-10 + 160)/(2)}}} or {{{x = (-10 - 160)/(2)}}} Break up the expression. 



{{{x = (150)/(2)}}} or {{{x =  (-170)/(2)}}} Combine like terms. 



{{{x = 75}}} or {{{x = -85}}} Simplify. 



So the possible solutions are {{{x = 75}}} or {{{x = -85}}} 

  

However, a negative length doesn't make sense. So ignore {{{x = -85}}}



So the only solution is {{{x = 75}}} which means that the width is 75 ft and the length is 85 ft.



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