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?
5^2 + 12^2 = 13^2
That is, x^2 + y^2 = 13
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
To determine the sum of the squares of two numbers using the given identity, let's derive the equation from the given information.
Let the two numbers be x and y.
Given:
1) Difference of the squares of the numbers: x^2 - y^2 = 5
2) Product of the numbers: xy = 6
We can rewrite the identity as:
(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2
Substitute the given values into the identity:
(x^2 + y^2)^2 = (5)^2 + (2xy)^2
(x^2 + y^2)^2 = 25 + (2xy)^2
Substitute the value of the product (xy = 6):
(x^2 + y^2)^2 = 25 + (2(6))^2
(x^2 + y^2)^2 = 25 + 36
(x^2 + y^2)^2 = 61
Taking the square root of both sides:
x^2 + y^2 = √61
Therefore, the sum of the squares of the two numbers is equal to √61, not 169.