The condition that x^3 - p x^2 + q x - r = 0 | vedclass

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

Question

(A) Given the cubic equation $x^3 - p x^2 + q x - r = 0$ ... $(i)$.Let the roots be $\alpha, \beta, \gamma$.From Vieta's formulas:$\alpha + \beta + \g

Vedclass · Accepted Answer

$r = pq$

Vedclass · Answer

(A) Given the cubic equation $x^3 - p x^2 + q x - r = 0$ ... $(i)$.Let the roots be $\alpha, \beta, \gamma$.From Vieta's formulas:$\alpha + \beta + \gamma = p$ ... $(ii)$$\alpha \beta + \beta \gamma + \gamma \alpha = q$ ... $(iii)$$\alpha \beta \gamma = r$ ... $(iv)$Given that two roots are equal in magnitude but opposite in sign,let $\alpha = -\beta$.Substituting $\alpha = -\beta$ into $(ii)$:$-\beta + \beta + \gamma = p \implies \gamma = p$.Substituting $\gamma = p$ into $(iv)$:$\alpha \beta (p) = r \implies -\beta^2 p = r \implies \beta^2 = -\frac{r}{p}$.Substituting $\alpha = -\beta$ and $\gamma = p$ into $(iii)$:$-\beta^2 + \beta p - \beta p = q \implies -\beta^2 = q$.Equating the two expressions for $\beta^2$:$-\frac{r}{p} = -q \implies r = pq$.

The condition that $x^3 - p x^2 + q x - r = 0$ may have two of its roots equal to each other but of opposite sign is

Similar Questions

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

Let $\alpha$ and $\beta$ be the roots of $x^2-6x-2=0$,with $\alpha > \beta$. If $a_n = \alpha^n - \beta^n$ for $n \geq 1$,then the value of $\frac{a_{10}-2a_8}{2a_9}$ is

If $\alpha+\beta=-2$ and $\alpha^3+\beta^3=-56$,then the quadratic equation whose roots are $\alpha$ and $\beta$ is

The harmonic mean of the roots of the equation $(5 + \sqrt{2})x^2 - (4 + \sqrt{5})x + 8 + 2\sqrt{5} = 0$ is:

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

Vedclass Test Series

Exam Paper Generator

Online Exam Module