Question 764740
WITHOUT MEMORIZED FORMULAS OR NAMES:
The function {{{p(x) = -4x^2 + 400x - 1000}}} can be re-written in a form that gives you a better understanding (and the answer to the problem.
{{{p(x) = -4x^2 + 400x - 1000}}} --> {{{p(x) = -4x^2 +400x-10000+900)}}} --> {{{p(x)=-4(x^2-100x+2500)+9000}}} --> {{{p(x)=-4(x-50)^2+9000}}}
When {{{highlight(x=50)}}} the firt term is zero {{{-4(x-50)^2=0}}}
and {{{p(x)=p(50)=9000}}}.
For any other value of {{{x}}}, {{{-4(x-50)^2<0}}} and {{{p(x)<9000}}},
so {{{highlight(x=50)}}} is the number of items sold that maximizes the profit,
and {{{p(50)=9000}}} is the maximum profit possible.
 
WITH WORDS AND FORMULAS TO MEMORIZE:
{{{p(x) = -4x^2 + 400x - 1000}}} is a {{{highlight(quadratic)}}} function (meaning a polynomial of degree 2).
A quadratic function, like {{{p(x) = -4x^2 + 400x - 1000}}}{{{graph(300,300,-20,100,-1000,10000,-4x^2 + 400x - 1000)}}}
graphs as a {{{highlight(parabola)}}} and has a maximum or a minimum (the {{{highlight(vertex)}}} of the parabola).
If the leading coefficient is negative, the function has a maximum. Otherwise, it's a minimum.
Quadratic functions are of the form {{{f(x)=ax^2+bx+c}}}
(or {{{f(x)=a(x-h)^2+k}}} when in vertex form).
The number you want to find is {{{highlight(h=-b/2a)}}}, the x-coordinate of the vertex of the parabola, which is part of
{{{x=h}}}, the equation of the axis of symmetry of the parabola.
If your teacher insist on memorization of formulas, you may have to memorize {{{h=-b/2a}}}, and for your problem you would write
{{{a=4}}}, {{{b=-400}}} --> {{{h=-(-400)/(2*4)}}} --> {{{h=50}}}