AP EAMCET · Maths · Quadratic Equation

Which of the following condition imply that roots of the equation \(\left(\frac{1}{4}\right) x^2+b x+c=0\) are integers?

A \(b^2-c>0\)
B \(b\) and \(c\) are even integers
C \(b^2-c\) is the square of an integer and \(b\) is an integer
D \(b\) and \(c\) are integers

Verified Solution

Answer & Solution

Correct Answer

(C) \(b^2-c\) is the square of an integer and \(b\) is an integer

Step-by-step Solution

Detailed explanation

We have, \(\frac{1}{4} x^2+b x+c=0\) By using quadratic formula \(x=\frac{-b \pm \sqrt{b^2-4 \times \frac{1}{4} \times c}}{2 \times \frac{1}{4}}\) \(x=\frac{-b \pm \sqrt{b^2-c}}{\frac{1}{2}}\) The roots are integer iff \(b\) is an integer and \(b^2-c\) is perfect square.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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