If the roots of x^4-10x^3+37x^2-60x+36=0 are | vedclass

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

Question

(A) Given equation: $x^4-10x^3+37x^2-60x+36=0$. Since the roots are $\alpha, \alpha, \beta, \beta$,by Vieta's formulas: Sum of roots: $2\alpha + 2\bet

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(A) Given equation: $x^4-10x^3+37x^2-60x+36=0$. Since the roots are $\alpha, \alpha, \beta, \beta$,by Vieta's formulas: Sum of roots: $2\alpha + 2\beta = 10 \Rightarrow \alpha + \beta = 5$ $(i)$. Product of roots: $\alpha^2\beta^2 = 36 \Rightarrow \alpha\beta = 6$ $(ii)$. From $(i)$,$\beta = 5 - \alpha$. Substituting into $(ii)$: $\alpha(5 - \alpha) = 6 \Rightarrow \alpha^2 - 5\alpha + 6 = 0$. Solving for $\alpha$: $(\alpha - 2)(\alpha - 3) = 0$,so $\alpha = 2$ or $\alpha = 3$. Since $\alpha Now,calculate $2\alpha + 3\beta - 2\alpha\beta$: $2(2) + 3(3) - 2(2)(3) = 4 + 9 - 12 = 1$.

If the roots of $x^4-10x^3+37x^2-60x+36=0$ are $\alpha, \alpha, \beta, \beta$ with $\alpha < \beta$,then find the value of $2\alpha+3\beta-2\alpha\beta$.

Similar Questions

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^4-8x^3+11x^2+32x-60=0$ and $\alpha < \beta < \gamma < \delta$, then $4\alpha+3\beta+2\gamma+\delta=$

If $(x-2)$ is a common factor of the expressions $x^2+ax+b$ and $x^2+cx+d$,then $\frac{b-d}{c-a}$ is equal to

What is the product of the roots of the equation $(x - 2)^2 - 3(x - 2) + 2 = 0$?

If $3 + 4i$ is a root of the equation ${x^2} + px + q = 0$ where $p$ and $q$ are real numbers,then:

The number of real roots of $|x|^2-5|x|+6=0$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module