If the difference between two positive numbers is 2 and the difference between their...
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If the difference between two positive numbers is 2 and the difference between their squares is 84, what is the sum of the two numbers?
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The difference of squares formula states that a² - b² = (a - b)(a + b). Given the difference (a - b) and the difference of squares (a² - b²), the sum (a + b) is found by dividing the difference of squares by the difference of the numbers.
Step-by-step Derivation:
Step 1: Let the two positive numbers be x and y, where x > y.
Step 2: According to the problem, the difference between the numbers is 2: x - y = 2.
Step 3: The difference between their squares is 84: x² - y² = 84.
Step 4: Use the algebraic identity for the difference of squares: x² - y² = (x - y)(x + y).
Step 5: Substitute the known values into the identity: 84 = (2)(x + y).
Step 6: Solve for (x + y) by dividing both sides by 2: x + y = 84 / 2 = 42.
Step 7: The sum of the two numbers is 42.