Question

Let d and e be two roots of the equation x2 + 2x + 2 = 0, then d15 + e15 is equal to

Answer 1

Answer:

d^15+e^15= - 256

Step-by-step explanation:

Given x²+2x+2=0

solving by Quadratic formula

we get

$x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\x=\frac{-2+\sqrt{2^2-4\cdot \:1\cdot \:2}}{2\cdot \:1}:\quad -1+i\\x=\frac{-2-\sqrt{2^2-4\cdot \:1\cdot \:2}}{2\cdot \:1}:\quad -1-i$

roots d=(-1+i) and e=(-1-i)

Now find

$d^{15}+e^{15}\\\left(-1+i\right)^{15}+\left(-1-i\right)^{15}\\\left(-1+i\right)^{15}:\quad -128-128i\\\left(-1-i\right)^{15}:\quad -128+128i\\=\left(-128-128i\right)+\left(-128+128i\right)\\=-256$

Answer 2

Answer:

(-1 + ι)¹⁵ + (-1 - ι)¹⁵

Step-by-step explanation:

Let d and e be two roots of the equation x2 + 2x + 2 = 0, then d15 + e15 is equal to

x² + 2x + 2 = 0

d & e are roots

d + e = - 2  ( sum of roots)

de = 2  (product of Roots)

roots =  (- 2 ± √(4-8))/2

= -1 ± ι

d = -1 + ι

e = -1 - ι

d¹⁵ + e¹⁵ = (-1 + ι)¹⁵ + (-1 - ι)¹⁵