Divide 32 into two parts such that the sum of the square of the parts is 674. What is the value of the parts?

Let the two parts be x and 32−x.
According to the question, the sum of their squares is 674:

x
2
+(32−x)
2
=674
x
2
+(1024−64x+x
2
)=674
2x
2
−64x+1024−674=0
2x
2
−64x+350=0
Divide the entire equation by 2:

x
2
−32x+175=0
Factorize the quadratic equation:

x
2
−25x−7x+175=0
x(x−25)−7(x−25)=0
(x−7)(x−25)=0
So, x=7 or x=25.
Thus, the two parts are 25 and 7.

Alternative Method (Using Options):
Check option C:
Sum of parts = 25+7=32
Sum of squares = 25
2
+7
2
=625+49=674. This matches perfectly.