SOLUTION: Two numbers have a difference of 24 . Find the numbers if the result of adding their sum and their product is a minimum. Can you please help me out? Thanks so much in advance:)

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Question 788876: Two numbers have a difference of 24 . Find the numbers if the result of adding their sum and their product is a minimum.
Can you please help me out? Thanks so much in advance:)
Can you also please show the steps it would really help me understand:) Thank-you
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
x= the smallest of the two numbers
x%2B24= the other number
x%2Bx%2B24= their sum
x%28x%2B24%29= their product
y=%28x%2Bx%2B24%29%2B%28x%28x%2B24%29%29= 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=%28x%2Bx%2B24%29%2B%28x%28x%2B24%29%29
y=%282x%2B24%29%2B%28x%5E2%2B24x%29

y=x%5E2%2B26x%2B169-169%2B24
y=%28x%2B13%29%5E2-169%2B24
y=%28x%2B13%29%5E2-145
As %28x%2B13%29%5E2%3E0 for all x%3C%3E-13 and %28x%2B13%29%5E2=0 for x=-13,
y=%28x%2B13%29%5E2-145%3E-145 for all x%3C%3E-13 and y=%28x-13%29%5E2-145=-145 is the minimum y for highlight%28x=-13%29
So x=highlight%28-13%29 and x%2B24=-13%2B24=highlight%2811%29 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%5E2%2Bbx%2Bc as general formula for the quadratic function, and
x=-b%2F2a as the formula to find the x-coordinate of the minimum.
In this case, a=1, b=26, and
x=-26%2F%282%2A1%29=-26%2F2=highlight%28-13%29 is the x-coordinate of the minimum.
The two numbers are x=highlight%28-13%29 and x%2B24=-13%2B24=highlight%2811%29