Question

If one root of the cubic equation $x^3-30x+133=0$ is $\frac{7+3\sqrt{3}i}{2}$. Find the real root of the cubic equation.

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• A. -7
• B. -7.00
• C. -7.0
• D. -Seven
• E. -seven

Answer 1

Answer: (A), (B), (C), (D), (E)

Let $\alpha ,\beta$ and $\gamma$ are the roots of the equation $x^3-30x+133=0$

If one root is $\alpha =\frac{7+3\sqrt{3}i}{2}$
Since, Imaginary roots of a polynomial equation with real coefficients, if exist, occurs in conjugate pairs.

$\therefore$ second root root is $\beta =\frac{7-3\sqrt{3}i}{2}$

Using relation between roots and coefficients,
Sum of the root = $\alpha +\beta +\gamma =\frac{-b}{a}=0$

$\frac{7+3\sqrt{3}i}{2}+\frac{7-3\sqrt{3}i}{2}+\gamma =0$

$\Rightarrow \gamma =-7$,
So, real root is -7