Question

The difference between the squares of two numbers is 120. The square of the smaller number is twice the greater number. Find the numbers.​

Answer 1

Let the numbers be x and y where x>y so,

x^2 - y^2 = 120.....(1)

y^2 = 2 x.....(2)

Putting (2) in (1), we get

x^2 - 2x = 120

x^2 - 2x - 120 = 0

x^2 - 12x + 10x - 120 = 0

x(x -12) +10(x - 12) = 0

(x-12)(x+10) = 0

So x can be 12 or -10, but we are given that the square of smaller number is twice the larger number, therefore the square of any number cant be negative,

Thus x cant take value of -10

So x = 12, so y^2 = 2 (12)

y^2 = 24

So y = √24 or - √24

y = 2√6 or -2√6

Hence the numbers are 12 and 2√6 or 12 and -2√6

Answer 2

Answer:

let the numbers be x and y , x >y

according to question,

x² - y² = 120 .......i

and y² = 2x........ii

putting the value of y² in ( i)

x² - 2x = 120

=> x ² - 2x - 120 =0

=> x² - 12x + 10 x - 120 =0

=> x( x - 12) + 10( x - 12) = 0

=> ( x - 12)( x + 10 ) = 0

=> x = 12 or x = - 10

now, rejecting -ve value of x and

taking x = 12 then

y = √2*12 = √24 = ± 2√6

,

so the numbers are 12 and 2√6

or 12 and -2√6