Question 466614: Find 2 numbers whose difference is 16 and whose product is minimun
Found 2 solutions by Edwin McCravy, Gogonati:
Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website!
Let x be one number
Let z be the other number
Their difference is |x-z|
|x-z| = 16
Case 1: Case 2:
x-z = 16 or x-z=-16
-z = 16-x -z=-16-x
z = -16+x z=16+x
z = x-16 z=x+16
Let y be their product zx
y = zx
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In case 1:
y = x(x-16)
y = x²-16x
Vertex formula theorem:
y = ax+bx+c has a minimum value when x = -b/a if a > 0
y = ax+bx+c has a maximum value when x = -b/a if a < 0
y = x²-16x
or
y = 1x²-16x+0
a = 1, b = -16, c = 0
y = 1x-16x+0 has a minimum value when x = -(-16)/1 = 16 since a = 1 > 0
z = 16-16 = 0
So the numbers are 16 and 0
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In case 2:
y = x(x+16)
y = x²+16x
Vertex formula theorem:
y = ax+bx+c has a minimum value when x = -b/a if a > 0
y = ax+bx+c has a maximum value when x = -b/a if a < 0
y = x²+16x
or
y = 1x²+16x+0
a = 1, b = 16, c = 0
y = 1x+16x+0 has a minimum value when x = -16/1 = -16 since a = 1 > 0
z = x+16 = -16+16 = 0
So the numbers are 0 and -16
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So there are two possibilities,
The two numbers could be 16 and 0 or else they could be 0 and -16
Edwin
Answer by Gogonati(855)  (Show Source):
You can put this solution on YOUR website! Let = x the first number, then the other number is x+16, and their product is the
function f(x)=x(x+16). The graph of this function is an upward parabola, where the
y-coordinate of its vertex is the minimum value of the function.
Write the equation of the parabola in standard form:
 , from the final equation we see that the vertex is:
(-8, -64). Therefore two numbers are -8 and -8+16=8, where their difference is
8-(-8)=16, and their minimum product is -64.
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