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The product of two consecutive positive integers is 306. Find the smallest integer.
Let the smallest integer be x, then the largest integer will be x + 1.
x(x + 1) = 306
x2 + x - 306 = 0
(x−17)(x + 18) = 0
x = 17, x = −18
The sum of the square of two positive integers is 208. If the square of the larger number is 18 times the smaller number, find the smaller numbers.
Let the smaller number be x. Square of the larger number is 18x.
Sum of their squares is
x2 + 18x = 208
(x + 26)(x−8) = 0 x = −26 or x = 8 … both are positive numbers
Find two consecutive odd positive integers the sum of whose squares is 290.
Let x be an odd positive integer.
Then an odd positive integer just greater than x is x + 2.
So,
x2 + (x + 2)2 = 290
x2 + 2x −143 = 0
(x + 13)(x - 11) = 0
x = −13 or x = 11 (integers are positive)
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