WBJEE · Maths · Quadratic Equation

If \(\alpha\) and \(\beta\) are roots of \(a x^{2}+b x+c=0,\) then
the equation whose roots are \(\alpha^{2}\) and \(\beta^{2},\) is

A \(a^{2} x^{2}-\left(b^{2}-2 a c\right) x+c^{2}=0\)
B \(a^{2} x^{2}+\left(b^{2}-a c\right) x+c^{2}=0\)
C \(a^{2} x^{2}+\left(b^{2}+a c\right) x+c^{2}=0\)
D \(a^{2} x^{2}+\left(b^{2}+2 a c\right) x+c^{2}=0\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(a^{2} x^{2}-\left(b^{2}-2 a c\right) x+c^{2}=0\)

Step-by-step Solution

Detailed explanation

Given. \(\alpha\) and \(\beta\) are roots of \(a x^{2}+b x+c=0\) \(\therefore \quad \alpha+\beta=\frac{-b}{a}\) and \(\quad a \beta=\frac{c}{a}\) Now, if the roots are \(\alpha^{2}\) and \(\beta^{2}\), then…

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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If \(\alpha\) and \(\beta\) are roots of \(a x^{2}+b x+c=0,\) then the equation whose roots are \(\alpha^{2}\) and \(\beta^{2},\) is

Step-by-step Solution

If \(\alpha\) and \(\beta\) are roots of \(a x^{2}+b x+c=0,\) then
the equation whose roots are \(\alpha^{2}\) and \(\beta^{2},\) is