$x=\frac{-(-1) \pm \sqrt{(-1)^2-4(1)(1)}}{2(1)}
Answer (2)
The given equation is simplified to find complex solutions through the quadratic formula, leading to the two solutions x = (1 ± i√3)/2. The calculations show that the discriminant is negative, indicating complex results. The final answers are the two complex numbers derived using the imaginary unit i.
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Simplify the expression under the square root.
Rewrite the equation using the simplified square root.
Express the square root of a negative number using the imaginary unit i .
Separate the expression into two complex solutions: x = 2 1 ± i 3
Explanation
Understanding the Problem We are given the equation x = 2 ( 1 ) − ( − 1 ) ± ( − 1 ) 2 − 4 ( 1 ) ( 1 ) Our goal is to simplify this expression and find the values of x .
Simplifying the Square Root First, let's simplify the expression under the square root: ( − 1 ) 2 − 4 ( 1 ) ( 1 ) = 1 − 4 = − 3 So the equation becomes x = 2 1 ± − 3
Introducing the Imaginary Unit Now, let's express the square root of -3 using the imaginary unit i , where i = − 1 .
− 3 = 3 × − 1 = 3 × − 1 = 3 i Substitute this back into the equation: x = 2 1 ± 3 i
Finding the Solutions Finally, we can separate the expression into two solutions: x 1 = 2 1 + 3 i x 2 = 2 1 − 3 i These are the two complex solutions for x .
Final Answer Therefore, the solutions to the equation are x 1 = 2 1 + 3 i , x 2 = 2 1 − 3 i x = 2 1 ± i 3
Examples
Complex numbers might seem abstract, but they're incredibly useful in electrical engineering. Imagine designing circuits where you need to analyze alternating currents. These currents can be represented using complex numbers, making calculations much easier. For example, the impedance of a circuit, which is a measure of its opposition to current flow, is often expressed as a complex number. By solving equations involving complex numbers, engineers can optimize circuit designs for efficiency and stability.
