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Question
There are three consecutive positive integers such that the sum of the square of the first and the product of other two is 154. What are the integers?
Solution
Let the first integer = x
then second integer = x + 1
and third integer = x + 2
Now according to the condition,
x2 + (x + 1)(x + 2) = 154
⇒ x2 + x2 + 3x + 2 - 154 = 0
⇒ 2x2 + 3x - 152 = 0
⇒ 2x2 + 19x - 16x - 152 = 0
⇒ x(2x + 19) - 8(2x + 19) = 0
⇒ (2x + 19)(x - 8) = 0
Either 2x + 19 = 0,
then 2x = -19
⇒ x = `-(19)/(2)`
But it is not possible as it is not an positive integer.
or
x - 8 = 0,
then x = 8
∴ Numbers are 8, (8 + 1) - 9 and (8 + 2) = 10.
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