Let P be the product

Let x be the first number

Let y be the second number.

x - y = 8 ---- (1) Pxy ----- (2)

Solve for x from equation (1): x = y + 8     -----> I understand this

Sub y + 8 for x into equation (2): P = (y+8)y = y2 + 8y -> I understand

Rewrite P = y2 + 8y in the standard form by completing the square. -->I understand

Vertex = (-4, -16). ---> I don't know how they found the vertex?

The product is minimum (P = −16) when y = −4. ---> I don't understand

Therefore x = y + 8 = − 4 + 8 = 4. ---> I understand

The two numbers are -4 and 4.

Can you thoroughly explain this example to me?

Expert Answers

An illustration of the letter 'A' in a speech bubbles


Let x and y be the two numbers

We want x-y = 8 and xy=P as small as possible

As you wrote, x=8+y

Once you have that, you can substitute into the other equation, and get

P=(y+8)y = y^2 + 8y

Just as you had.





NOW:

the nice thing about this, is you have P written as a function of only 1 variable, and P is a parabola.  Parabolas come in two flavors: rightsideup and upsidedown:


"Right side up" parabolas have a minimum.  There is a spot on the parabola that is lower than any other.  (The "vertex") But they don't have a maximum.  That is, you can always go higher and higher.

"Up side down" parabolas have a maximum.  There is a spot on the parabola that is higher than any other (the vertext).  But they don't have a minimum.  You can always go lower and lower.  

Our parabola looks like:





We want to find that spot at the bottom (the red thing is a makeshift arrow).  That is where P is at its lowest






`P=ay^2+by+c = 1 y^2 + 8y + 0`

One way to find the vertex is to use the formula:

`(-b)/(2a)`

This gives you the first coordinate of the vertex.  In our case

`(-8)/(2*1) = -4`

To get the second coordinate, plug the first one into the parabola equation.  In our case:

`P = (-4)^2 + 8 (-4) = -16`

So the vertext is (-4,-16)

So, when y = -4, then P = -16 (and x = y+8 = 4)





Completing the square is a little harder, but it is another way to find the vertex.

Completing the square means you want to write the equation in the form:

`P = a(y-h)^2 + k`

The nice thing about writing P this way is that the vertex is (h,k)

That is, if you have the equation written in standard form, you can tell at a glance what the vertex is.

If you have that done, great.  Just in case:



We want:

`y^2 + 8y = a(y-h)^2 + k`

The "a" is the same in both cases.  That is, for us,

`P=ay^2+by+c = 1 y^2 + 8y + 0`

The "a" here is 1, so we have:

`P = 1(y-h)^2 + k`


So, we want:


`y^2 + 8y = (y-h)^2 + k`

Expand the right hand side:

`y^2 + 8y = (y-h)(y-h) + k = y^2 - yh - yh + h^2 + k`


So:

`y^2 + 8y + 0 =  y^2 - (2h)y + h^2 + k`

We want the two sides to be equal.

So we need -2h = 8, and we need  `0=h^2+k`

So we need h = -4

If h=-4, then `0=h^2+k=(-4)^2+k = 16+k`

So we need k=-16


So, if we plug these back into what we started with, we have:

`y^2 + 8y = (y-h)^2 + k`

`y^2 + 8y = (y+4)^2 -16`


This is standard form.  The vertex is (h,k) which is (-4,-16)
 (you take the negative of what is in parentheses)

That means, the minimum possible P is -16, and it happens when y=-4
(which means x must be 4)

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial