The sum of two positive numbers is 14. What is the smallest possible value of the sum of their squares? How do you know this is the smallest sum? (Show how you set up and arrive at the answer)
¢+U=14
v = 14 - T
S=22 + y?
$=¢'+(14-212
S=22+196-282+12
S = 222 _ 28r + 196
S' = 47 - 28
0 = 4x - 28
42 = 28
X =7
y = 14 - 7
y = 7
S = 2(7)2 - 28(7) + 196
S = 98 - 196 + 196
S = 98
To find the smallest possible value of the sum of their squares, we can use the equation S = 2x^2 - 28x + 196, where x is one of the positive numbers.
Since the sum of the two numbers is 14, we can express the other positive number as y = 14 - x.
Substituting y into the equation, we get S = 2x^2 - 28x + 196 - 28(14 - x).
Simplifying the equation, we have S = 2x^2 - 28x + 196 - 392 + 28x.
Combining like terms, S = 2x^2 - 196.
To find the smallest possible value of S, we need to find the vertex of the quadratic equation. The x-coordinate of the vertex can be found using the formula x = -b/2a.
In this case, a = 2 and b = 0, so x = -0/2(2) = 0.
Substituting x = 0 back into the equation S = 2x^2 - 196, we find that the smallest possible value of S is 2(0)^2 - 196 = -196.
Therefore, the smallest possible value of the sum of their squares is -196. This is the smallest sum because any other positive value of x would result in a larger positive value for S.