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Mathematics

Find two consecutive positive odd numbers, the sum of whose squares is 74.

Quadratic Equations

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Answer

Let two consecutive positive odd numbers be x and (x + 2).

⇒ (x)2 + (x + 2)2 = 74

⇒ x2 + x2 + 4 + 4x = 74

⇒ 2x2 + 4x + 4 - 74 = 0

⇒ 2x2 + 4x - 70 = 0

⇒ 2(x2 + 2x - 35) = 0

⇒ x2 + 2x - 35 = 0

⇒ x2 + 7x - 5x - 35 = 0

⇒ x(x + 7) - 5(x + 7) = 0

⇒ (x + 7)(x - 5) = 0

⇒ (x + 7) = 0 or (x - 5) = 0

⇒ x = -7 or x = 5.

Since, numbers are positive and even,

∴ x ≠ -7.

∴ x = 5 and (x + 2) = 7.

Hence, numbers are 5, 7.

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