The sum of the squares of three consecutive natural numbers as 149. Find the numbers
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Solution
Let the numbers be x, x + 1 and x + 2 according to the given hypothesis.
𝑥2 + (𝑥 + 1)2 + (𝑥 + 2)2 = 149
⇒ 𝑥2 + 𝑥2 + 1 + 2𝑥 + 𝑥2 + 4 + 4𝑥 = 149
⇒ 3𝑥2 + 6𝑥 + 5 - 149 = 0
⇒ 3𝑥2 + 𝑥 - 144 = 0
⇒ 𝑥2 + 2𝑥 - 48 = 0
⇒ 𝑥(𝑥 + 8) - 6(𝑥 + 8) = 0
⇒ (𝑥 + 8)(𝑥 - 6) = 0
⇒ x = -8 or x = 6
Considering the positive value of x
𝑥 = 6, 𝑥 + 1 = 7 𝑎𝑛𝑑 𝑥 + 2 = 8
∴ The three consecutive numbers are 6, 7, 8.
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