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You probably first encountered complex numbers when you studied values of (called roots or zeros) for which the following equation is satisfied:
For (as we will assume), this equation may be written as
Let's denote the second-degree polynomial on the left-hand side of this equation by :
This is called a monic polynomial because the coefficient of the highest-power term is 1. When looking for solutions to the quadratic equation , we are really looking for roots (or zeros) of the polynomial . The fundamental theorem of algebra says that there are two such roots. When wehave found them, we may factor the polynomial as follows:
In this equation, z 1 and z 2 are the roots we seek. The factored form shows clearly that , meaning that the quadratic equation is solved for and . In the process of factoring the polynomial , we solve the quadratic equation and vice versa.
By equating the coefficients of , and z 0 on the left-and right-hand sides of [link] , we find that the sum and the product of the roots and obey the equations
You should always check your solutions with these equations.
Completing the Square. In order to solve the quadratic equation (or, equivalently, to find the roots of the polynomial , we “complete the square” on the left-hand side of [link] :
This equation may be rewritten as
We may take the square root of each side to find the solutions
In the equation that defines the roots and , the term iscritical because it determines the nature of the solutions for and . In fact, we may define three classes of solutions depending on .
(i) Overdamped . In this case, the roots and are
These two roots are real, and they are located symmetrically about the point . When , they are located symmetrically about 0 at the points . (In this case, ) Typical solutions are illustrated in [link] .
(ii) Critically Damped . In this case, the roots and are equal (we say they are repeated):
These solutions are illustrated in [link] .
(iii) Underdamped . The underdamped case is, by far, the most fascinating case. When , then the square root in the solutions for and ( [link] ) produces an imaginary number. We may write as and write as
These complex roots are illustrated in [link] . Note that the roots are
purely imaginary when , producing the result
In this underdamped case, the roots and are complex conjugates:
Thus the polynomial also takes the form
and are related to the original coefficients of the polynomial as follows:
Always check these equations.
Let's explore these connections further by using the polar representations for and :
Then [link] for the polynomial may be written in the “standard form”
[link] is now
These equations may be used to locate
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