Question 788876
{{{x}}}= the smallest of the two numbers
{{{x+24}}}= the other number
{{{x+x+24}}}= their sum
{{{x(x+24)}}}= their product
{{{y=(x+x+24)+(x(x+24))}}}= the result of adding their sum and their product.
We need to find the value of {{{x}}} that minimizes the y value for that function.
 
Let's simplify that expression:
{{{y=(x+x+24)+(x(x+24))}}}
{{{y=(2x+24)+(x^2+24x)}}}
{{{y=2x+24+x^2+24x}}
{{{y=x^2+26x+24}}
That is a quadratic function with a minimum. (The graph is a parabola).
To find the coordinates of the minimum we can apply formulas taught in class, or we can just complete the square.
 
COMPLETING THE SQUARE:
No need to memorize formulas (unless your teacher thinks you should).
{{{y=x^2+26x+24}}}
{{{y=x^2+26x+169-169+24}}}
{{{y=(x+13)^2-169+24}}}
{{{y=(x+13)^2-145}}}
As {{{(x+13)^2>0}}} for all {{{x<>-13}}} and {{{(x+13)^2=0}}} for {{{x=-13}}},
{{{y=(x+13)^2-145>-145}}} for all {{{x<>-13}}} and {{{y=(x-13)^2-145=-145}}} is the minimum {{{y}}} for {{{highlight(x=-13)}}}
So {{{x=highlight(-13)}}} and {{{x+24=-13+24=highlight(11)}}} are the two numbers we were looking for.
 
WITH FORMULAS:
THe formulas you were given may be slightly different.
You were probably given {{{y=ax^2+bx+c}}} as general formula for the quadratic function, and
{{{x=-b/2a}}} as the formula to find the x-coordinate of the minimum.
In this case, {{{a=1}}}, {{{b=26}}}, and
{{{x=-26/(2*1)=-26/2=highlight(-13)}}} is the x-coordinate of the minimum.
The two numbers are {{{x=highlight(-13)}}} and {{{x+24=-13+24=highlight(11)}}}