Mathematics
The sum of the squares of two consecutive natural numbers is 41. Find the numbers.
Quadratic Equations
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Answer
Let two consecutive natural numbers = x and x + 1.
⇒ x2 + (x + 1)2 = 41
⇒ x2 + x2 + 1 + 2x = 41
⇒ 2x2 + 2x + 1 - 41 = 0
⇒ 2x2 + 2x - 40 = 0
⇒ 2(x2 + x - 20) = 0
⇒ x2 + x - 20 = 0
⇒ x2 + 5x - 4x - 20 = 0
⇒ x(x + 5) - 4(x + 5) = 0
⇒ (x - 4)(x + 5) = 0
⇒ (x - 4) = 0 or (x + 5) = 0
⇒ x = 4 or x = -5.
Since, x is a natural number so, x ≠ -5.
∴ x = 4 and (x + 1) = 5.
Hence, numbers = 4, 5.
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