The sum of 2 positive numbers is 151. the lesser number is 19 more than the square root of the greater number. what is the value of the greater number minus the lesser number?

Let the two positive number be x and y.

Assume

From the given condition: The sum of 2 positive number is 151 and the lesser number is 19 more than the square root of the greater number.

then we have,  

    .....[1]

Substitute the value of in equation [1],

Subtract 19 from both the sides, we get

on simplify:

or

squaring both the sides, we get

Using Identities   on right hand side of above expression:

or

or we can write it as:

By, solving above quadratic equation we get,

x =121

to find the value of y:

Substitute the value of x in above equation to get value for y:

⇒ y= 30

The value of greatest number minus the  lesser would be x-y, i.e,  121-30 = 91