How to Calculate Two Numbers Using Their Sum and Difference of Squares
Given:
- \( S \): the sum of two numbers (\( x + y \))
- \( D \): the difference of their squares (\( x^2 - y^2 \))
Steps:
-
Set up relationships:
\( x + y = S, \quad x^2 - y^2 = D \)
-
Use the difference of squares formula:
\( x^2 - y^2 = (x + y)(x - y) \)
Substituting \( S \):
\( D = S \cdot (x - y) \)
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Solve for \( x - y \):
\( x - y = \frac{D}{S} \)
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Find \( x \) and \( y \) using the sum and difference formulas:
\( x = \frac{(x + y) + (x - y)}{2} \)
\( y = \frac{(x + y) - (x - y)}{2} \)
Examples
Example 1: The sum of two numbers is 24, and the difference of their squares is 144. What are the numbers?
Solution:
1. Calculate \( x - y \):
\( x - y = \frac{144}{24} = 6 \)
2. Calculate \( x \) and \( y \):
\( x = \frac{24 + 6}{2} = 15 \)
\( y = \frac{24 - 6}{2} = 9 \)
Result: The numbers are 15 and 9.
Example 2: The sum of two numbers is 31, and the difference of their squares is 465. What are the numbers?
Solution:
1. Calculate \( x - y \):
\( x - y = \frac{465}{31} = 15 \)
2. Calculate \( x \) and \( y \):
\( x = \frac{31 + 15}{2} = 23 \)
\( y = \frac{31 - 15}{2} = 8 \)
Result: The numbers are 23 and 8.
Example 3: The sum of two numbers is 45, and the difference of their squares is 675. What are the numbers?
Solution:
1. Calculate \( x - y \):
\( x - y = \frac{675}{45} = 15 \)
2. Calculate \( x \) and \( y \):
\( x = \frac{45 + 15}{2} = 30 \)
\( y = \frac{45 - 15}{2} = 15 \)
Result: The numbers are 30 and 15.