Which equation has roots that are the squares | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

Vedclass · Accepted Answer

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

Vedclass · Answer

(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 $ax^2 + bx + c = 0$?

Similar Questions

The harmonic mean of two numbers is $-\frac{8}{5}$ and their geometric mean is $2$. The quadratic equation whose roots are twice those numbers is

The sum of the roots of a quadratic equation is $2$ and the sum of their cubes is $98$. Then the equation is:

If $\alpha$ and $\beta$ are the roots of $9x^2 + 6x + 1 = 0$,then the equation with the roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is

If $\alpha, \beta, \gamma$ are the roots of $2x^3 - 2x - 1 = 0$,then $(\Sigma \alpha \beta)^2$ is equal to

Let $\alpha, \beta$ be the roots of the equation $x^2-ax-b=0$ with $\operatorname{Im}(\alpha) < \operatorname{Im}(\beta)$. Let $P_n=\alpha^n-\beta^n$. If $P_3=-5 \sqrt{7} i, P_4=-3 \sqrt{7} i, P_5=11 \sqrt{7} i$ and $P_6=45 \sqrt{7} i$,then $|\alpha^4+\beta^4|$ is equal to . . . . . .

Vedclass Test Series

Exam Paper Generator

Online Exam Module