Animated Solution for Mathematics - Complex Numbers: Let $\alpha$ and $\beta$ be the roots of the equation $x^2+(2i-1)=0$. Then, the value of $|{\alpha }^8+{\beta }^8|$ is equal to :

We are given the quadratic equation $x^2+(2i-1)=0$. We need to determine the value of $|{\alpha }^8+{\beta }^8|$, where $\alpha$ and $\beta$ are the roots of this equation.

Many students instinctively reach for the quadratic formula, but this often leads to unnecessary complexity. Instead, we should focus on the structural properties of the equation.

By rearranging the given equation, we isolate the squared term:

Since $\alpha$ and $\beta$ are the roots of the equation, they must satisfy this relationship. Therefore, we have:

Notice the symmetry here: both roots yield the same value when squared. This observation allows us to bypass the need to calculate the individual roots $\alpha$ and $\beta$.

To find ${\alpha }^8$ and ${\beta }^8$, we raise our squared expression to the fourth power. Since ${\alpha }^2=1-2i$, it follows that:

Applying the same logic to $\beta$, we find:

We now calculate the modulus of the sum:

Using the properties of the modulus, specifically $|z^n|=|z{|}^n$ and $|cz|=|c||z|$, we simplify the expression:

Next, we calculate the modulus of the complex number $1-2i$:

Substituting the value of the modulus back into our expression, we get:

Since $(\sqrt{5}{)}^4=25$, the final calculation is:

By identifying the underlying symmetry and utilizing the properties of complex numbers, we have determined that the final answer is 50.