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

(A) Given the quadratic equation $ax^2 + bx + c = 0$,the sum of the roots is $\alpha + \beta = -\frac{b}{a}$ and the product of the roots is $\alpha \

Vedclass · Accepted Answer

$\frac{c(a - b)}{a^2}$

Vedclass · Answer

(A) Given the quadratic equation $ax^2 + bx + c = 0$,the sum of the roots is $\alpha + \beta = -\frac{b}{a}$ and the product of the roots is $\alpha \beta = \frac{c}{a}$.We need to evaluate the expression $\alpha \beta^2 + \alpha^2 \beta + \alpha \beta$.Factoring out $\alpha \beta$,we get:$\alpha \beta (\beta + \alpha) + \alpha \beta = \alpha \beta (\alpha + \beta + 1)$.Substituting the values of $\alpha + \beta$ and $\alpha \beta$:$= \frac{c}{a} \left( -\frac{b}{a} + 1 \right) = \frac{c}{a} \left( \frac{a - b}{a} \right) = \frac{c(a - b)}{a^2}$.

If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c = 0$,then what is the value of $\alpha \beta^2 + \alpha^2 \beta + \alpha \beta$?

Similar Questions

Let $\alpha$ and $\beta$ be the roots of $x^{2}-6x-2=0$. If $a_{n}=\alpha^{n}-\beta^{n}$ for $n \geq 1$,then the value of $\frac{a_{10}-2a_{8}}{3a_{9}}$ is ..... .

If the ratio of the roots of the equations $x^2 + bx + c = 0$ and $x^2 + qx + r = 0$ are equal,then:

If $\lambda$ is the ratio of the roots of the quadratic equation in $x$,$3m^2x^2 + m(m - 4)x + 2 = 0$,then the least value of $m$ for which $\lambda + \frac{1}{\lambda} = 1$ is:

If the roots of the equation $5x^2 - 7x + k = 0$ are reciprocal to each other,then the value of $k$ is:

If the roots of the quadratic equation $\frac{x - m}{mx + 1} = \frac{x + n}{nx + 1}$ are reciprocal to each other,then

Vedclass Test Series

Exam Paper Generator

Online Exam Module