If 2 + 3i is one of the roots of the equation | vedclass

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

Question

(B) Given the equation $2x^3 - 9x^2 + kx - 13 = 0$ with real coefficients,complex roots must occur in conjugate pairs.Since $2 + 3i$ is a root,its con

Vedclass · Accepted Answer

exists and is equal to $\frac{1}{2}.$

Vedclass · Answer

(B) Given the equation $2x^3 - 9x^2 + kx - 13 = 0$ with real coefficients,complex roots must occur in conjugate pairs.Since $2 + 3i$ is a root,its conjugate $2 - 3i$ must also be a root.Let the roots be $\alpha = 2 + 3i,$ $\beta = 2 - 3i,$ and $\gamma$ be the real root.According to the relationship between roots and coefficients for a cubic equation $ax^3 + bx^2 + cx + d = 0,$ the product of the roots is given by $\alpha \beta \gamma = -\frac{d}{a}.$Here,$a = 2$ and $d = -13,$ so the product of the roots is $\alpha \beta \gamma = -\frac{-13}{2} = \frac{13}{2}.$Calculating the product of the complex roots: $\alpha \beta = (2 + 3i)(2 - 3i) = 2^2 - (3i)^2 = 4 + 9 = 13.$Substituting these into the product formula: $13 \cdot \gamma = \frac{13}{2}.$Solving for $\gamma,$ we get $\gamma = \frac{1}{2}.$Thus,the real root exists and is equal to $\frac{1}{2}.$

If $2 + 3i$ is one of the roots of the equation $2x^3 - 9x^2 + kx - 13 = 0,$ where $k \in R,$ then the real root of this equation

Similar Questions

If $x = \sqrt{5} + 2$,then the value of $\frac{2x^2 - 3x - 2}{3x^2 - 4x - 3}$ is equal to

Solve the given two equations and select the correct option.
$I.$ $77 x^{2}+58 x+8=0$
$II.$ $42 y^{2}+59 y+20=0$

Solve the given equations and choose the correct option.
$I.$ $x = \sqrt{(36)^{1/2} \times (1296)^{1/4}}$
$II.$ $2y + 3z = 33$
$III.$ $6y + 5z = 71$

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

Solve the given two equations and select the correct answer from the given options.
$I.$ $821 x^{2} - 757 x^{2} = 256$
$II.$ $\sqrt{196} y^{3} - 12 y^{3} = 16$

Vedclass Test Series

Exam Paper Generator

Online Exam Module