If alpha and beta are the roots of the equati | vedclass

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

Question

(B) Since $\alpha$ and $\beta$ are roots of $ax^2 + bx + c = 0$,we have $a\alpha^2 + b\alpha + c = 0$ and $a\beta^2 + b\beta + c = 0$.From $a\alpha^2

Vedclass · Accepted Answer

$\frac{b^2 - 2ac}{a^2c^2}$

Vedclass · Answer

(B) Since $\alpha$ and $\beta$ are roots of $ax^2 + bx + c = 0$,we have $a\alpha^2 + b\alpha + c = 0$ and $a\beta^2 + b\beta + c = 0$.From $a\alpha^2 + b\alpha + c = 0$,we get $\alpha(a\alpha + b) = -c$,so $(a\alpha + b) = -\frac{c}{\alpha}$.Similarly,$(a\beta + b) = -\frac{c}{\beta}$.Substituting these into the expression:$\frac{1}{(a\alpha + b)^2} + \frac{1}{(a\beta + b)^2} = \frac{1}{(-c/\alpha)^2} + \frac{1}{(-c/\beta)^2} = \frac{\alpha^2}{c^2} + \frac{\beta^2}{c^2} = \frac{\alpha^2 + \beta^2}{c^2}$.Using the identity $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$,where $\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$:$\frac{(\alpha + \beta)^2 - 2\alpha\beta}{c^2} = \frac{(-b/a)^2 - 2(c/a)}{c^2} = \frac{b^2/a^2 - 2c/a}{c^2} = \frac{b^2 - 2ac}{a^2c^2}$.

If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c = 0$,then what is the value of $\frac{1}{(a\alpha + b)^2} + \frac{1}{(a\beta + b)^2}$?

Similar Questions

If $\alpha$ and $\beta$ are roots of the equation $x^2 + px + \frac{3p}{4} = 0$ such that $|\alpha - \beta| = \sqrt{10}$,then $p$ belongs to the set

If $\alpha, \beta$, where $\alpha < \beta$, are the roots of the equation $\lambda x^{2} - (\lambda + 3)x + 3 = 0$ such that $\frac{1}{\alpha} - \frac{1}{\beta} = \frac{1}{3}$, then the sum of all possible values of $\lambda$ is:

If $\alpha$ and $\beta$ are the roots of $x^2-10x-8=0$ with $\alpha > \beta$,and $a_n = \alpha^n - \beta^n$ for $n \in N$,then the value of $\frac{a_{10}-8a_8}{5a_9}$ is:

If the ratio of the roots of $x^2 + bx + c = 0$ and $x^2 + qx + r = 0$ is the same,then

The condition that one root of the equation $ax^2 + bx + c = 0$ is three times the other is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module