Question 383959
To find the max or min for a quadratic equation, convert to vertex form ({{{y=a(x-h)^2+k}}}) by completing the square.
{{{P= -2x^2 + 20x-2}}}
{{{P=-2(x^2-10x)-2}}}
{{{P=-2(x^2-10x+25)-2+2(25)}}}
{{{P=-2(x-5)^2-2+50}}}
{{{P=-2(x-5)^2+48}}}
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The vertex is ({{{5}}},{{{48}}}), with a maximum profit of {{{48}}} hundred dollars which occurs when {{{x=5}}} thousand units are produced.
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{{{drawing(300,300,-2,8,-5,55,grid(1),circle(5,48,0.25),blue(line(5,-100,5,100)),graph(300,300,-2,8,-5,55,0,-2(x-5)^2+48))}}}