How to Solve Quadratic Equations with Complex Numbers: A Step-by-Step Guide
Introduction
In this article, we'll explore a specific quadratic equation and provide a detailed step-by-step solution. Specifically, we'll solve the equation z^2 - 1 - iz - 1 0, where the term 1_i is interpreted as 1 - i. This involves using the quadratic formula and converting complex numbers into polar form to find the roots.
Solving the Quadratic Equation
Let's start with the given equation:
z^2 - 1 - iz - 1 0.
To solve this, we can use the quadratic formula, which is given by:
z u00BD(-b u00B1 u221A(b^2 - 4ac))
First, we need to identify the coefficients a, b, and c. In our equation:
a 1 b -i - 1 c -1Now, we can plug these into the quadratic formula:
z u00BD(-(-i - 1) u00B1 u221A((-i - 1)^2 - 4 u00D7 1 u00D7 (-1)))
First, let's simplify the discriminant:
(-i - 1)^2 - 4 u00D7 1 u00D7 (-1) (i^2 2i 1) 4
Given that i^2 -1, we get:
(-1 2i 1) 4 4i 4
Now we have:
z u00BD(1 i u00B1 u221A(4i 4))
Using Polar Form to Find the Square Root
To simplify further, we can express the term 4i 4 in polar form. Polar form is given by:
r u221A(a^2 b^2)
where a and b are the real and imaginary parts of the complex number, respectively.
r u221A(4^2 4^2) u221A(16 16) u221A32 4u221A2
The angle u03B8 is given by:
u03B8 arctan(b/a)
Since our term is 4i 4, we have:
u03B8 arctan(4/4) arctan(1) u03C0/4
The square root of a complex number in polar form is given by:
u221A(r)(cos(u03B8/2) u2212 isin(u03B8/2))
Plugging in the values, we get:
u221A(4u221A2)(cos(u03C0/8) u2212 isin(u03C0/8))
This simplifies to approximately:
u221A(4u221A2)(0.924 - 0.383i) u2248 2.11 u2212 0.89i
The other root is thus:
-2.11 0.89i
Final Roots of the Equation
Now, we can substitute these back into our original formula:
z u00BD(1 i u00B1 2.11 u2212 0.89i)
This gives us two roots:
Root 1:
z u00BD(1 i 2.11 u2212 0.89i) u2248 u00BD(3.11 0.11i) u2248 1.555 0.055i
Root 2:
z u00BD(1 i u2212 2.11 0.89i) u2248 u00BD(-1.11 1.89i) u2248 -0.555 0.945i
Thus, the roots of the equation z^2 - 1 - iz - 1 0 are approximately:
1.555 0.055i -0.555 0.945iThis detailed guide should help you understand how to solve similar quadratic equations with complex numbers and the importance of converting terms into polar form to simplify the solution.
Conclusion
Solving quadratic equations with complex numbers can be a bit tricky, but with a solid understanding of the quadratic formula and the ability to convert complex numbers into polar form, the process becomes much clearer. This article provided a step-by-step solution and important insights into the process. Keep practicing, and you'll be able to tackle more complex equations with ease!