Question

Use the identity (x^2+y^2)^2=(x^2−y^2)^2+(2xy)^2 to determine the sum of the squares of two numbers if the difference of the squares of the numbers is 5 and the product of the numbers is 6.

Is it 169?

Answer 1

5^2 + 12^2 = 13^2
That is, x^2 + y^2 = 13

Answer 2

The difference of the squares of the numbers is 5 mean:

x² - y² = 5

The product of the numbers is 6 mean:

x ∙ y = 6

Replace this values in equation:

( x² + y² )² = ( x² − y² )² + ( 2 x y )²

( x² + y² )² = 5² + ( 2 ∙ 6 )²

( x² + y² )² = 25 + 12²

( x² + y² )² = 25 + 144

( x² + y² )² = 169

Take square root of both sides:

x² + y² = √169

x² + y² = 13