vikasp99 wrote:
The sum of two numbers is 13, and their product is 30. What is the sum of the squares of the two numbers?
(A) –229
(B) –109
(C) 139
(D) 109
(E) 229
We can let the numbers be a and b. Thus:
a + b = 13 and ab = 30
Since a = 13 - b, we can substitute 13 - b in the equation ab = 30 and we have:
(13 - b)b = 30
13b - b^2 = 30
b^2 - 13b + 30 = 0
(b - 10)(b - 3) = 0
b = 10 or b = 3
Notice that when b = 10, a = 3, and when b = 3, a = 10. Therefore, the two numbers that have a sum of 13 and a product of 39 are 3 and 10, and the sum of their squares is is 9 + 100 = 109.
Alternate Solution:
Let’s square each side of a + b = 13:
(a + b)^2 = 169
a^2 + 2ab + b^2 = 169
Since ab = 30, we can substitute 2ab = 60 in the last equality:
a^2 + b^2 + 60 = 169
a^2 + b^2 = 109
Answer: D
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