Animated Solution for Mathematics - Complex Numbers: Let $\alpha$ and $\beta$ be two roots of the equation $x^2+2x+2=0$, then ${\alpha }^{15}+{\beta }^{15}$ is equal to :

Imagine you are standing before the quadratic equation $x^2+2x+2=0$. At first glance, it looks like a standard, unassuming problem.

However, as an aspirant, you know that appearances can be deceiving. If you dive straight into the quadratic formula, you will find yourself wading through a sea of calculations. There is a more elegant path, a shortcut that reveals the hidden symmetry of the complex plane.

Instead of brute force, let us look at the structure. We can rewrite the equation as $x^2+2x+1+1=0$.

Because $x^2+2x+1$ is a perfect square, this transforms our equation into:

This simplifies to $(x+1{)}^2=-1$. Suddenly, the path clears as we step into the realm of complex numbers. Taking the square root of both sides, we find $x+1=\pm i$, which gives us our roots:

Now, the challenge is to find ${\alpha }^{15}+{\beta }^{15}$. If you try to expand $(-1+i{)}^{15}$ using the binomial theorem, you will be trapped in a labyrinth of terms.

Let us be smarter and square these roots first. For $\alpha$, we have:

Since $i^2=-1$, the real parts cancel out, leaving us with ${\alpha }^2=-2i$. Similarly, for $\beta$:

This is the turning point. Squaring has simplified our complex numbers into pure imaginary numbers. Geometrically, you have just doubled the angle of these roots on the Argand plane and squared their magnitudes.

We need the fifteenth power, but we have the squares. Let us aim for the fourteenth power first:

This expands to $(-2{)}^7\cdot i^7$. We know that $(-2{)}^7=-128$, and since $i^4=1$, we have $i^7=i^4\cdot i^3=1\cdot (-i)=-i$.

Thus, ${\alpha }^{14}=-128\cdot (-i)=128i$. By the same logic:

We are almost at the finish line. To find ${\alpha }^{15}$, we simply multiply ${\alpha }^{14}$ by $\alpha$:

Similarly, for ${\beta }^{15}$:

Now, for the grand finale, we add them together:

The imaginary terms $128i$ and $-128i$ vanish into thin air. This leaves us with the final result:

This is the beauty of mathematics. By choosing the right perspective—by seeing the structure rather than just the numbers—we turned a daunting calculation into a graceful dance of cancellation.