Which equation has roots that are the squares | vedclass

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(C) Let the roots of the equation $ax^2 + bx + c = 0$ be $\alpha$ and $\beta$.Then,$\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$.We

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$a^2x^2 - (b^2 - 2ac)x + c^2 = 0$

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(C) Let the roots of the equation $ax^2 + bx + c = 0$ be $\alpha$ and $\beta$.Then,$\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$.We want to find an equation whose roots are $\alpha^2$ and $\beta^2$.The sum of the new roots is $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = (-\frac{b}{a})^2 - 2(\frac{c}{a}) = \frac{b^2}{a^2} - \frac{2c}{a} = \frac{b^2 - 2ac}{a^2}$.The product of the new roots is $\alpha^2\beta^2 = (\alpha\beta)^2 = (\frac{c}{a})^2 = \frac{c^2}{a^2}$.The required quadratic equation is $x^2 - (	ext{sum of roots})x + (	ext{product of roots}) = 0$.Substituting the values,we get $x^2 - (\frac{b^2 - 2ac}{a^2})x + \frac{c^2}{a^2} = 0$.Multiplying by $a^2$,we get $a^2x^2 - (b^2 - 2ac)x + c^2 = 0$.

Which equation has roots that are the squares of the roots of the equation $ax^2 + bx + c = 0$?

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