```Question 181935

We basically have this triangle set up:

{{{drawing(500,500,-0.5,2,-0.5,3.2,
line(0,0,0,3),
line(0,3,2,0),
line(2,0,0,0),
locate(-0.2,1.5,x),
locate(1,-0.2,x-7),
locate(1,2,13)
)}}}

Since the legs are {{{x}}} and {{{x-7}}} this means that {{{a=x}}} and {{{b=x-7}}}

Also, since the hypotenuse is {{{13}}}, this means that {{{c=13}}}.

{{{x^2+(x-7)^2=13^2}}} Plug in {{{a=x}}}, {{{b=x-7}}}, {{{c=13}}}

{{{x^2+(x-7)^2=169}}} Square {{{13}}} to get {{{169}}}.

{{{x^2+x^2-14x+49=169}}} FOIL

{{{x^2+x^2-14x+49-169=0}}} Subtract 169 from both sides.

{{{2x^2-14x-120=0}}} Combine like terms.

Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=2}}}, {{{b=-14}}}, and {{{c=-120}}}

Let's use the quadratic formula to solve for x

{{{x = (-(-14) +- sqrt( (-14)^2-4(2)(-120) ))/(2(2))}}} Plug in  {{{a=2}}}, {{{b=-14}}}, and {{{c=-120}}}

{{{x = (14 +- sqrt( (-14)^2-4(2)(-120) ))/(2(2))}}} Negate {{{-14}}} to get {{{14}}}.

{{{x = (14 +- sqrt( 196-4(2)(-120) ))/(2(2))}}} Square {{{-14}}} to get {{{196}}}.

{{{x = (14 +- sqrt( 196--960 ))/(2(2))}}} Multiply {{{4(2)(-120)}}} to get {{{-960}}}

{{{x = (14 +- sqrt( 196+960 ))/(2(2))}}} Rewrite {{{sqrt(196--960)}}} as {{{sqrt(196+960)}}}

{{{x = (14 +- sqrt( 1156 ))/(2(2))}}} Add {{{196}}} to {{{960}}} to get {{{1156}}}

{{{x = (14 +- sqrt( 1156 ))/(4)}}} Multiply {{{2}}} and {{{2}}} to get {{{4}}}.

{{{x = (14 +- 34)/(4)}}} Take the square root of {{{1156}}} to get {{{34}}}.

{{{x = (14 + 34)/(4)}}} or {{{x = (14 - 34)/(4)}}} Break up the expression.

{{{x = (48)/(4)}}} or {{{x =  (-20)/(4)}}} Combine like terms.

{{{x = 12}}} or {{{x = -5}}} Simplify.

So the possible answers are {{{x = 12}}} or {{{x = -5}}}

However, a negative length is NOT possible. So {{{x = -5}}} is NOT a solution

So the only answer is {{{x = 12}}}

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