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Question
Find two consecutive integers such that the sum of their squares is 61
Solution
Let first integer = x
Then second integer = x + 1
According to the condition,
(x)2 + (x + 1)2 = 61
⇒ x2 + x2 + 2x + 1 = 61
⇒ 2x2 + 2x + 1 - 61 = 0
⇒ 2x2 + 2x - 60 = 0
⇒ x2 + x - 30 = 0 ...(Dividing by 2)
⇒ x2 + 6x - 5x - 30 = 0
⇒ x(x + 6) -5(x + 6) = 0
⇒ (x + 6)(x - 5) = 0
Either x + 6 = 0,
then x = -6
or
x - 5 = 0,
then x = 5
(i) If x = -6, then
First integer = -6
and second = -6 + 1 = -5
(ii) If x = 5, then
First integer = 5
and second = 5 + 1 = 6
∴ Required integers are (-6, -5),(5, 6).
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