How to Calculate Two Numbers Using Their Difference and Difference of Squares
Given:
- \( D \): the difference of two numbers (\( x - y \))
- \( S \): the difference of their squares (\( x^2 - y^2 \))
Steps:
-
Set up relationships:
\( x - y = D, \quad x^2 - y^2 = S \)
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Use the difference of squares formula:
\( x^2 - y^2 = (x + y)(x - y) \)
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Solve for \( x + y \):
\( x + y = \frac{S}{D} \)
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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: Two numbers have a difference of 15, and the difference of their squares is 465. What are the numbers?
Solution:
1. Calculate \( x + y \):
\( x + y = \frac{465}{15} = 31 \)
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 2: Two numbers have a difference of 9, and the difference of their squares is 981. What are the numbers?
Solution:
1. Calculate \( x + y \):
\( x + y = \frac{981}{9} = 109 \)
2. Calculate \( x \) and \( y \):
\( x = \frac{109 + 9}{2} = 59 \)
\( y = \frac{109 - 9}{2} = 50 \)
Result: The numbers are 59 and 50.
Example 3: Two numbers have a difference of 8, and the difference of their squares is 160. What are the numbers?
Solution:
1. Calculate \( x + y \):
\( x + y = \frac{160}{8} = 20 \)
2. Calculate \( x \) and \( y \):
\( x = \frac{20 + 8}{2} = 14 \)
\( y = \frac{20 - 8}{2} = 6 \)
Result: The numbers are 14 and 6.