Two numbers have a difference of 16. The result of adding their sum and their product is a minimum.Find the #s

Question

This needs to be completed by completing the square. When I tried this, I got (8,8). But the answer is supposed to be (7,-9).

Answer 1

The answer isn't really all that hard once you figure out what it's asking. The trick is simply to set up your equations properly:

1. Let the two numbers be x and y, where x > y.

2. "Two numbers have a difference of 16." This statement simply means that x - y = 16.

3. "The result of adding their sum (x + y) and their product (x*y) is a minimum." This statement means that (x + y) + (x*y) = c, where c is some minimum value. NOTE THAT "MINIMUM VALUE" DOES NOT IMPLY THAT C = 0!!!

So here are the required equations:

Eq. 1: x - y = 16
Eq. 2: x + xy + y = c

So there are 2 equations with 3 unknowns (x, y, c). We will have to find the minimum value of c in order to complete the system. Eq. 1 can be rewritten as:

x = y + 16

Substituting this into Eq. 2:

(y + 16) + (y + 16)*y + y = c

Combining terms and re-arranging provides:

y² + 18y + 16 = c

This is a quadratic equation. It can be restated as:

y² + 18 + (16 - c) = 0

Using the quadratic formula:

y = (-18 ± sqrt(18² - 4(1)(16-c)))/(2*1)

("sqrt" means "square root")

So in order for y to be a real number, all of the terms inside the sqrt() must be equal to or greater than zero. This provides the 3rd equation to the system:

Eq. 3: 18² - 4(16 - c) >= 0

(>= means "greater than or equal to")

Eq. 3 can be re-arranged to get an expression for c:

18² >= 4(16 - c)

c >= 16 - 18²/4

c >= -65

Thus, c must be greater than or equal to -65. Returning to the problem statement, the requirement is to use the minimum value for c, which is -65. Note that any value below -65 will result in an imaginary (non-real) result for y. Hence, Eq. 2 becomes:

Eq. 2: x + xy + y = -65

Again substituting Eq. 1 into Eq. 2 provides:

(y + 16) + (y + 16)y + y = -65

y² + 18y + 16 = -65

y² + 18y + 81 = 0

You can now complete the square to solve for y:

(y + a)(y + b) = 0

Notice that y will only have exactly one value, so a = b, and this becomes:

(y + a)(y + a) = (y + a)² = 0

Therefore a² = 81 and 2*a = 18. Obviously, a = 9. This means that:

(y + 9)² = 0

So y must equal -9. Substituting this back into Eq. 1 provides x = 7. Thus, the solution is (7, -9).

You can also solve this expression with the quadratic formula (I find it simpler to use this method):

y = (-18 ± sqrt(18² - 4(1)(49)))/(2*1)

y = (-18 ± sqrt(0))/(2*1)

y = -18/2 = -9

Finally, we substitute this value back into Eq. 1 to obtain:

x = 16 + y = 16 - 9 = 7

Therefore, x = 7 and y = -9. Again, the solution is (7, -9)

Answer 2

Let the two numbers be x and y. Then

y - x = 16 or y = x + 16.

To minimize

(x+y) + xy, substitute y = x + 16 to get

2x + 16 + x(x + 16), or after rearranging

terms,

x^2 + 18x + 16 = (x+9)^2 - 65. The vertex of this quadratic is on

the line x = -9, and since it opens upward, the minimum occurs

at x = -9.

So x = -9 and y = 7 are the two numbers required.

Answer 3

x - y = 16 →y = x - 16
S = x + y + xy
= x + x - 16 + x(x - 16)
= 2x - 16 + x^2 - 16x
= x^2 - 14x - 16
dS/dx = 2x - 14
= 0 for stationary points
x = 7
Stationary point is a minimum (it is an upright parabola)
Thus the two numbers are 7 and -9

S = 7 + -9 + 7 x -9 = -65