z%5E2-(5-i)z%2B(2-2i)%3D0.jpeg "(2+i)z^2-(5-i)z+(2-2i)=0")
Solving the Quadratic Equation: (2 + i)z² - (5 - i)z + (2 - 2i) = 0
This article explores the solution process for the quadratic equation (2 + i)z² - (5 - i)z + (2 - 2i) = 0, where 'z' is a complex number.
Understanding Complex Numbers
Before delving into the solution, let's briefly understand complex numbers. A complex number has the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as √-1.
The Quadratic Formula
The quadratic formula is a fundamental tool for solving quadratic equations of the form ax² + bx + c = 0. The formula provides the solutions for 'x' as:
x = (-b ± √(b² - 4ac)) / 2a
Applying the Formula
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Identify Coefficients: In our equation, we have:
- a = (2 + i)
- b = -(5 - i)
- c = (2 - 2i)
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Substitute Values: Substitute these coefficients into the quadratic formula:
z = ( (5 - i) ± √((-5 + i)² - 4(2 + i)(2 - 2i)) ) / 2(2 + i)
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Simplify: Perform the necessary algebraic manipulations to simplify the expression.
- Simplify the square root: (-5 + i)² - 4(2 + i)(2 - 2i) = -23 - 18i
- Calculate the denominator: 2(2 + i) = 4 + 2i
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Solve for z:
- z = ((5 - i) ± √(-23 - 18i)) / (4 + 2i)
To simplify the square root of a complex number, we can use the following steps:
- Find the magnitude: |√(-23 - 18i)| = √(23² + 18²) = √845
- Find the angle: θ = arctan(-18/23)
- Express the square root in polar form: √(-23 - 18i) = √845(cos(θ) + i sin(θ))
Substitute this value back into the solution for 'z'.
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Rationalize the Denominator: To get a simplified solution, multiply the numerator and denominator by the conjugate of the denominator (4 - 2i).
Finding the Solutions
Following these steps will lead to two solutions for 'z'. The solutions will be in the form of complex numbers (a + bi).
Conclusion
Solving quadratic equations with complex coefficients requires careful application of the quadratic formula and complex number manipulation. By understanding the properties of complex numbers and applying the appropriate simplification techniques, we can find the solutions to such equations.