Question

divide 18 into two parts such that twice the sum of their squares is 5 times their product

Answer 1

let the numbers be x & y 

le the product be = xy 

the sum of squares will be x^2+y^2 

A/C to the question,
2(x²+y²)=5xy 

=> x²+y²=(5/2)xy 
=> x²+y²+2xy = (5/2)xy +2xy  ( adding 2xy both the sides)
=> (x+y)²=(9/2)xy 
=> 18²=(9/2)xy
 => 2 x 324/9 =xy 

=>  xy=72
=> x=72/y 
=> 72/y + y=18 ( as per question) 
=> 72+y²=18y
=> y²-18y+72=0
=> y²-12y-6y+72=0 ( splitted the middle term)
=> (y-12)-6(y-12)=0

so, y = 6 or can be 12 
hope it helps!! 

Answer 2

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