#### Question

The sum of the squares of three consecutive natural numbers as 149. Find the numbers

#### Solution

Let the numbers be x, x + 1 and x + 2 according to the given hypothesis.

𝑥^{2} + (𝑥 + 1)^{2} + (𝑥 + 2)^{2} = 149

⇒ 𝑥^{2} + 𝑥^{2} + 1 + 2𝑥 + 𝑥^{2} + 4 + 4𝑥 = 149

⇒ 3𝑥^{2} + 6𝑥 + 5 - 149 = 0

⇒ 3𝑥^{2} + 𝑥 - 144 = 0

⇒ 𝑥^{2} + 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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#### APPEARS IN

Solution The Sum of the Squares of Three Consecutive Natural Numbers as 149. Find the Numbers Concept: Solutions of Quadratic Equations by Factorization.