If one root of x^3-7x^2+36=0 is twice the oth | vedclass

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

Question

(B) Let the roots of the cubic equation $x^3-7x^2+0x+36=0$ be $\alpha, 2\alpha,$ and $\beta$.From the relation between roots and coefficients:Sum of r

Vedclass · Accepted Answer

$9$

Vedclass · Answer

(B) Let the roots of the cubic equation $x^3-7x^2+0x+36=0$ be $\alpha, 2\alpha,$ and $\beta$.From the relation between roots and coefficients:Sum of roots: $\alpha + 2\alpha + \beta = 7 \implies 3\alpha + \beta = 7 \implies \beta = 7 - 3\alpha$.Sum of roots taken two at a time: $\alpha(2\alpha) + 2\alpha\beta + \beta\alpha = 0 \implies 2\alpha^2 + 3\alpha\beta = 0$.Since $\alpha 
eq 0$ (as $36 
eq 0$),we divide by $\alpha$: $2\alpha + 3\beta = 0$.Substitute $\beta = 7 - 3\alpha$ into the equation: $2\alpha + 3(7 - 3\alpha) = 0$.$2\alpha + 21 - 9\alpha = 0 \implies -7\alpha = -21 \implies \alpha = 3$.Then the roots are $\alpha = 3$ and $2\alpha = 6$.The sum of these two roots is $3 + 6 = 9$.

If one root of $x^3-7x^2+36=0$ is twice the other,then the sum of those two roots is

Similar Questions

If $a, b$ and $c$ are the roots of $x^3+qx+r=0$,then $(a-b)^2+(b-c)^2+(c-a)^2=$ (in $q$)

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+px^2+qx+r=0$,then $(\alpha+\beta)(\beta+\gamma)(\gamma+\alpha)=$

If $\alpha, \beta$ are the roots of $a x^2+b x+c=0$,then the quadratic equation whose roots are $\sqrt{5} \alpha, \sqrt{5} \beta$ is

If one root of the quadratic equation $ax^2 + bx + c = 0$ is equal to the $n$th power of the other,then $(ac^n)^{1/(n+1)} + (a^nc)^{1/(n+1)} =$

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - 5x - 3 = 0$,then what is the equation whose roots are $\frac{1}{2\alpha - 3}$ and $\frac{1}{2\beta - 3}$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module