Question

What is the minimum product of two numbers whose difference is 48? What are the numbers?

The minimum product is_____
The two number yield this product?______
Simplify your answer

Answer 1

Hello,

Let's assume x the greatest number
y the smallest

x-y=48
P(x)=x*y=x*(x-48)=x²-2*24x+24²-24²=(x-24)²-576

Minimum if the vertex when x=24 and y=-24

Minimum product=-576
The two numbers are 24 and -24 but i don't know how to simplify!

Answer 2

Let x and y be the two numbers. If the difference of the numbers is 48,
y - x = 48
y = 48 + x
For the product of the numbers,
P = x y = x (48 + x) = 48x + x²
To determine the minimum product, differentiate the equation, equate to zero and solve for x.
dP = 0 = 48 + 2x ; x = -24
The value of y is equal to 24
Thus, the minimum product is -576 and the numbers are -24 and 24.