Question 1002047
h, hypotenuse
x and 30-x are lengths of the legs.
{{{h^2=x^2+(30-x)^2}}}
{{{h^2=x^2+30^2-60x+x^2}}}
{{{h^2=2x^2-60x+900}}}


Find the x value or values for the minimum {{{h^2}}}.
{{{2x^2-60x+900=0}}}
{{{x^2-30x+450=0}}}
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Discriminant, {{{900-4*450=NegativeNumber}}}
This method, finding the roots, will not work here because {{{h^2}}} as a function does not intersect the x-axis.


You can complete the square for making into standard form, and then read the vertex  (a minimum point) from that form of the equation.  The term to use is {{{(30/2)^2=15^2}}}, or 225.


{{{x^2-30x+15^2+450-225=0}}}
{{{(x-15)^2+225=0}}}
Showing x=15 for the vertex, and then one leg of the triangle is 15, and therefore the other leg is {{{30-x=30-15=15}}} also.