Numbers Finder by Sum of Squares and Difference of Squares

Enter the sum of squares and difference of squares to find two numbers instantly!

How to Calculate Two Numbers Using the Sum of Squares and Difference of Squares

Let \( x \) and \( y \) represent the two numbers. Suppose we know their sum of squares (\( S \)) and difference of squares (\( D \)).

Initial formulas: Sum of squares: \( x^2 + y^2 = S\) Difference of squares: \(x^2 - y^2 = D \)
Add the two equations to solve for \( x^2 \):

\( (x^2 + y^2) + (x^2 - y^2) = S + D \)

\( 2x^2 = S + D \)

\( x^2 = \frac{S + D}{2} \)
Subtract the two equations to solve for \( y^2 \):

\( (x^2 + y^2) - (x^2 - y^2) = S - D \)

\( 2y^2 = S - D \)

\( y^2 = \frac{S - D}{2} \)
Find \( x \) and \( y \): \( x = \sqrt{\frac{S + D}{2}} \) \( y = \sqrt{\frac{S - D}{2}} \)

Solution:

Calculate \( x \):

\( x = \sqrt{\frac{S + D}{2}} = \sqrt{\frac{74 + 24}{2}} = \sqrt{49} = 7 \)

Calculate \( y \):

\( y = \sqrt{\frac{S - D}{2}} = \sqrt{\frac{74 - 24}{2}}= \sqrt{25} = 5 \)

Result: The two numbers are 7 and 5.

Solution:

Calculate \( x \):

\( x = \sqrt{\frac{S + D}{2}} = \sqrt{\frac{313 + 25}{2}} = \sqrt{169} = 13 \)

Calculate \( y \):

\( y = \sqrt{\frac{S - D}{2}} = \sqrt{\frac{313 - 25}{2}}= \sqrt{144} = 12 \)

Result: The two numbers are 13 and 12.