Question

Let α,β be the roots of the quadratic equation  x²+√6x+3=0. Then  \(\frac{α^23+β^23 + α ^14 + β ^14 }{α ^15 + β ^15 + α 160 + β ^{10}}\)  is equal to

• A. 9
• B. 12
• C. 81
• D. 729

Hint:

When dealing with complex roots of quadratic equations, converting to polar form can be beneficial for simplification. Review trigonometric identities for eval uating cosine and sine of various angles. Pay careful attention to signs and pow ers when simplifying.

Answer 1

The Correct Option is C

Given: \[ \alpha, \beta = \frac{-\sqrt{6} \pm \sqrt{6 - 12}}{2}. \]

We can rewrite this as: \[ \alpha, \beta = \sqrt{3} e^{\pm 3\pi i / 4}. \]

Required Expression: \[ \left(\sqrt{3}\right)^{23} 2\cos\left(69\pi / 4\right) = \frac{-\sqrt{6} \pm \sqrt{6} i}{2} + \left(\sqrt{3}\right)^{14} 2\cos\left(42\pi / 4\right). \] 
Simplifying further: \[ \left(\sqrt{3}\right)^{15} 2\cos\left(45\pi / 4\right). \]

Additionally, we know: \[ \left(\sqrt{3}\right)^{10} 2\cos\left(30\pi / 4\right) \sqrt{3}^{8} = 81. \]