-x-(3%2Bi)-x-(3-i).jpeg "(x-1) X-(3+i) X-(3-i)")
Factoring and Solving the Equation: (x-1)x-(3+i)x-(3-i)
This article explores the process of factoring and solving the equation (x-1)x-(3+i)x-(3-i) = 0.
Understanding the Equation
The given equation is a quadratic equation with complex coefficients. It can be rewritten in a more standard form as:
x² - (4+i)x - (3-i) = 0
Factoring the Equation
To factor this equation, we need to find two numbers that:
- Multiply to give -(3-i)
- Add to give -(4+i)
This is similar to factoring a regular quadratic equation, but with the added complexity of dealing with complex numbers.
One approach is to use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
Where:
- a = 1
- b = -(4+i)
- c = -(3-i)
By substituting these values into the quadratic formula and simplifying, we can find the two roots of the equation, which are the factors of the expression.
Solving the Equation
The solutions to the equation, also known as its roots, are the values of x that make the equation true.
By factoring the equation as described above, we can find the two roots. Since the quadratic formula is used for finding the roots, we can directly get the solutions without explicitly factoring the expression.
Conclusion
The equation (x-1)x-(3+i)x-(3-i) = 0 is a quadratic equation with complex coefficients. Factoring this equation involves finding two numbers that satisfy specific conditions related to multiplication and addition. The solutions to the equation can be obtained using the quadratic formula, providing the two roots of the equation.