algebra question

Question

One number is 5 less than 4 times a second number. Find a pair of such numbers so that their product is as small as possible.

These two numbers are _____. (Use a comma to separate your numbers.)

The smallest possible product is_____

Answer 1

Let one number be "y", let the second number be "x". Then 4 times the second number is 4x and 5 less than that is 4x - 5 so y = 4x - 5.

The product of the two numbers is x•y or, plugging in "4x - 5" into this in place of "y", we get the product is (x)(4x - 5). Or we can say f(x) = (x)(4x - 5) or

f(x) = 4x² - 5x

We are trying to minimize this product. The shape of the quadratic is a parabola which looks like this:

So, we are trying to find the vertex. The vertex of the parabola f(x) = ax² + bx + c is found at the x-value of -b/(2a). In our case, a = 4 and b = -5 so x = -b/(2a) = 5/8. So the second number is 5/8. To find the other number, use y = 4x - 5 so y = 4(5/8) - 5 = -5/2.

So the two numbers are 5/8 and -5/2 and their product (which is the smallest product) is -25/16

Answer 2

Assume such numbers are positive integers then first number = 3 and second = 2

Then smallest product is 6