In this article, we will walk through the process of solving the quadratic equation involving complex numbers: z² + 2z – i = 0. The goal is to understand how to work with complex solutions and how to express them in polar form (using the angle and magnitude). If you’ve already attempted to solve the equation but got stuck at the polar form representation, this guide will help clarify the steps.
1. Expressing the Equation in Terms of Real and Imaginary Parts
To start solving the equation z² + 2z – i = 0, let’s assume that z is a complex number, z = x + iy, where x is the real part and y is the imaginary part. Substituting this into the original equation:
- z² = (x + iy)² = x² + 2ixy – y²
- 2z = 2(x + iy) = 2x + 2iy
- -i = -i
Now, we can substitute these into the equation:
(x² – y² + 2ixy) + 2x + 2iy – i = 0
From here, separate the real and imaginary parts:
- Real part: x² – y² + 2x = 0
- Imaginary part: 2xy + 2y – 1 = 0
We now have two equations that must be solved simultaneously.
2. Solving the System of Equations
We now need to solve the following system of equations:
- x² – y² + 2x = 0
- 2xy + 2y – 1 = 0
By solving these equations, we can find values for x and y. You can start by manipulating the second equation:
2y(x + 1) = 1
From this, solve for y:
y = 1 / [2(x + 1)]
Substitute this value of y into the first equation and solve for x. After finding the value of x, substitute it back to get y.
3. Converting to Polar Form
Once you have the complex number in rectangular form (z = x + iy), the next step is to convert it into polar form, which is often written as:
z = r(cos θ + i sin θ)
where r is the modulus (magnitude) of the complex number and θ is the argument (angle). The modulus is given by:
r = √(x² + y²)
The argument θ can be found using the formula:
θ = tan⁻¹(y/x)
After calculating r and θ, express the solution in polar form. In the given problem, you should arrive at the polar form expression:
z = -1 ± 2^(1/4)(cos(π/8) + i sin(π/8))
4. Conclusion
By following the above steps, you can solve the given quadratic equation with complex solutions. The key is to break the problem into real and imaginary parts, solve the system of equations, and then convert the final complex number into polar form. This process can be applied to similar problems involving complex numbers in quadratic equations.
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