Quadratic equations are getting me tired

Algebra Level 5

Consider an equation with p , q p, q real roots x 2 − x a + 1 b = 0 x^{2} - \frac{x}{a} + \frac{1}{b} = 0 .

Also 1 a , 1 b \frac{1}{a}, \frac{1}{b} are positive integers and 1 a ≥ 2. \frac{1}{a}\geq 2.

[ p q ] + [ q p ] = ( I n t e g e r ) 2 [\frac{p}{q}] + [\frac{q}{p}] = (Integer)^{2} , if and only if (where [.] denotes floor function)

both p p and q q need not be integers p p need not be an integer, q q is an integer p , q p, q are both integers p p is an integer, q q need not be an integer

Archit Tripathi by Archit Tripathi , 23, India

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1 solution

The equation x 2 − 20 x + 98 = 0 x^2-20x+98=0 satisfies the hypotheses of the problem and its roots p = 10 + 2 p=10+\sqrt{2} and q = 10 − 2 q=10-\sqrt{2} are not integer numbers.

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x 2 − 4 x + 1 x^2-4x+1 is another example.

Joe Mansley - 2 weeks, 1 day ago

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