# Quadratic equations and inequalities / applications of quadratic functions question

## Homework Statement

find two numbers whose sum is 20 and whose product is a maximum.

## Homework Equations

the first number is X
the second number is 20-x

3. The solution
X(20-X)=0
-X^2+ 20x=0
x=-b/2a = -20/2(-1) = 10
20 - x =20 -10 = 10

the numbers are 10 and 10

i just dont get why / how you know to put x and 20 - x and why you would use the axis of symmetry to find the numbers

and sry mods i posted originally in the wrong thread.

Last edited:

## Answers and Replies

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Integral
Staff Emeritus
Science Advisor
Gold Member
From the problem statement you have 2 numbers which sum to 20, that is x and 20-x.

It should be obvious that ( x )+ (20 -x) = 20 so you have represented the 2 numbers in general. Now you need to find when the product x(20-x) is a maximum.

Now if you were in a calculus class you would take the derivitive and set it to zero. Since you are not doing this I will have to assume that you are not in calculus. You have the problem of finding the maximum of the parabola, using properties of a parabola. The maximum will lie on the axis of symetry of the parabola, so all you need do is find the point on the parabola which lies on the symetry axis.

thanks
not in calc but next year trig then pre and then calc

HallsofIvy
Science Advisor
Homework Helper
X(20-X)=0 is not true. You have the function 20X- X2 which is a parabola with maximum value at its vertex. You can find the (X,Y) coordinates of the vertex by completing the square.