Question 183085
You're on the right track, you just need to solve the equation:



{{{(x/4)^2 + ((16-x)/4)^2 = 10 }}} Start with the given equation.



{{{x^2/16 + ((16-x)/4)^2 = 10 }}} Square {{{x/4}}} to get {{{(x/4)^2=(x/4)(x/4)=x^2/16}}}



{{{x^2/16 + (16-x)^2/16 = 10 }}} Square {{{(16-x)/4}}} to get {{{((16-x)/4)^2=((16-x)/4)((16-x)/4)=(16-x)^2/16}}}



{{{(x^2+ (16-x)^2)/16 = 10 }}} Combine the fractions.



{{{x^2+ (16-x)^2 = 10(16) }}} Multiply both sides by 16.



{{{x^2+ (16-x)^2 = 160 }}} Multiply



{{{x^2+ 256-32x+x^2 = 160 }}} FOIL



{{{x^2+ 256-32x+x^2 -160 =0}}} Subtract 160 from both sides.



{{{2x^2-32x+96=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=2}}}, {{{b=-32}}}, 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 = (-(-32) +- sqrt( (-32)^2-4(2)(96) ))/(2(2))}}} Plug in  {{{a=2}}}, {{{b=-32}}}, and {{{c=96}}}



{{{x = (32 +- sqrt( (-32)^2-4(2)(96) ))/(2(2))}}} Negate {{{-32}}} to get {{{32}}}. 



{{{x = (32 +- sqrt( 1024-4(2)(96) ))/(2(2))}}} Square {{{-32}}} to get {{{1024}}}. 



{{{x = (32 +- sqrt( 1024-768 ))/(2(2))}}} Multiply {{{4(2)(96)}}} to get {{{768}}}



{{{x = (32 +- sqrt( 256 ))/(2(2))}}} Subtract {{{768}}} from {{{1024}}} to get {{{256}}}



{{{x = (32 +- sqrt( 256 ))/(4)}}} Multiply {{{2}}} and {{{2}}} to get {{{4}}}. 



{{{x = (32 +- 16)/(4)}}} Take the square root of {{{256}}} to get {{{16}}}. 



{{{x = (32 + 16)/(4)}}} or {{{x = (32 - 16)/(4)}}} Break up the expression. 



{{{x = (48)/(4)}}} or {{{x =  (16)/(4)}}} Combine like terms. 



{{{x = 12}}} or {{{x = 4}}} Simplify. 



So the answers are {{{x = 12}}} or {{{x = 4}}} 



This means that the second length of the square is either 


{{{16-12=4}}} or {{{16-4=12}}}


Note: either way, the two side lengths are 12 and 4



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Answer:


So the length of the two pieces are 12 and 4 inches.