Out of three consecutive positive integers, the middle number is p. If three times the square of the largest is greater than the sum of the squares of the other two numbers by 67; calculate the value of p.
Quadratic Equations
54 Likes

Question

Out of three consecutive positive integers, the middle number is p. If three times the square of the largest is greater than the sum of the squares of the other two numbers by 67; calculate the value of p.

54 Likes

KnowledgeBoat · Accepted Answer

Let the numbers be (p - 1), p and (p + 1).

According to question,

⇒ 3(p + 1)² = (p - 1)² + p² + 67

⇒ 3(p² + 1 + 2p) = (p² + 1 - 2p) + p² + 67

⇒ 3p² + 3 + 6p = 2p² - 2p + 68

⇒ 3p² - 2p² + 6p + 2p + 3 - 68 = 0

⇒ p² + 8p - 65 = 0

⇒ p² + 13p - 5p - 65 = 0

⇒ p(p + 13) - 5(p + 13) = 0

⇒ (p - 5)(p + 13) = 0

⇒ p - 5 = 0 or p + 13 = 0

⇒ p = 5 or p = -13

Since, numbers are natural numbers,

∴ p ≠ -13.

∴ p - 1 = 4 and p + 1 = 6.

Hence, p = 5.