Unlocking the Curious Combination of Number Theory and Algebra: ISI Entrance Problem, TOMATO Subjective 17

Lets dive into a captivating problem that unites algebra and number theory, creating a compelling challenge. This puzzle revolves around a particular type of mathematical equation. The ultimate aim is to prove that the solutions to this equation cannot be expressed as straightforward fractions like 1/2 or 3/4.

Here is the question we solve:

If the coefficients of a quadratic equation

are all odd integers, show that the roots cannot be rational.

To solve this problem we use the quadratic formula and a clever concept known as "parity check." At the heart of the matter is demonstrating that the discriminant, b² - 4ac, doesn't fit the definition of a "perfect square."

Now, let's delve into the parity check. Regardless of the interplay between even and odd numbers, the left side of the equation consistently maintains an even quality, while the right side remains steadfastly odd. This leads to a conclusion: the solutions do not exist

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