Animated Solution for Mathematics - Complex Numbers: If the equation, $x^2+bx+45=0$ $(b\in \mathbf{R})$ has conjugate complex roots and they satisfy $|z+1|=2\sqrt{10}$, then :

We are given the quadratic equation $x^2+bx+45=0$ with real coefficients and complex roots. Because the coefficients are real, the roots must exist as a conjugate pair.

Let the roots be $z=p+iq$ and $\bar{z}=p-iq$, where $p,q\in \mathbb{R}$ and $q eq 0$.

According to Vieta's formulas, the sum of the roots is $z+\bar{z}=-b$. Substituting our definitions, we get:

The product of the roots is $z\cdot \bar{z}=45$. Expanding this product yields: This relation, $p^2+q^2=45$, serves as our primary algebraic constraint.

We are given the condition $|z+1|=2\sqrt{10}$. Substituting $z=p+iq$ into this expression, we have:

Using the definition of the modulus $|x+iy|=\sqrt{x^2+y^2}$, we square both sides to obtain:

Expanding the geometric equation, we get:

We can substitute our "golden key" $p^2+q^2=45$ into this equation:

Since we previously established that $2p=-b$, it follows that $-b=-6$, which means $b=6$.

To find the final value requested, we evaluate $b^2-b$:

The final result is 30.