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Magazine Chapter 3: Problem 102
Solve the equations over the complex numbers. \(x^{2}-2 x+4=0\)
Short Answer
Expert verified
The roots are \(1 + i\sqrt{3}\) and \(1 - i\sqrt{3}\).
Step by step solution
01
Identify the Coefficients
In the quadratic equation \(x^2 - 2x + 4 = 0\), identify the coefficients as follows:\(a = 1\), \(b = -2\), and \(c = 4\). These values will be used in the quadratic formula.
02
Write the Quadratic Formula
The quadratic formula to find the roots of the equation \(ax^2 + bx + c = 0\) is given by: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
03
Calculate the Discriminant
Compute the discriminant \(b^2 - 4ac\) using the coefficients: \((-2)^2 - 4 \cdot 1 \cdot 4 = 4 - 16 = -12\). The negative discriminant indicates that the roots are complex numbers.
04
Solve Using the Quadratic Formula
Substitute the known values into the quadratic formula: \(x = \frac{-(-2) \pm \sqrt{-12}}{2 \times 1}\). This simplifies to \(x = \frac{2 \pm \sqrt{-12}}{2}\).
05
Simplify the Square Root
The term \(\sqrt{-12}\) can be written as \(\sqrt{-1 \times 12} = \sqrt{-1} \cdot \sqrt{12} = i \cdot 2\sqrt{3}\). Thus, \(\sqrt{-12} = 2i\sqrt{3}\).
06
Simplify the Expression
Plug \(\sqrt{-12} = 2i\sqrt{3}\) back into the quadratic formula: \(x = \frac{2 \pm 2i\sqrt{3}}{2}\). This simplifies to \(x = 1 \pm i\sqrt{3}\).
07
Conclusion
The roots of the equation are complex numbers: \(1 + i\sqrt{3}\) and \(1 - i\sqrt{3}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
The quadratic formula is a powerful tool used to find the roots of a quadratic equation. A quadratic equation is any expression in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients. The formula used to solve this type of equation is:
- \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Discriminant
The discriminant is a key part of the quadratic formula that helps determine the nature of the roots of a quadratic equation. The discriminant is given by the expression \( b^2 - 4ac \). By evaluating the discriminant:
- If it is positive, the equation has two distinct real roots.
- If it is zero, the equation has exactly one real root (a repeated root).
- If it is negative, the equation has two complex roots.
Complex Roots
Complex roots emerge from quadratic equations with a negative discriminant. When this happens, we handle the square root of a negative number, introducing an imaginary unit \( i \). The complex roots of the equation are calculated as:
- \( x = 1 + i\sqrt{3} \)
- \( x = 1 - i\sqrt{3} \)
Imaginary Numbers
Imaginary numbers are an extension of the real numbers, allowing us to perform operations involving the square roots of negative numbers. The basic imaginary unit is \( i \), defined as \( i = \sqrt{-1} \). So, when computing square roots for the discriminant in our exercise:
- \( \sqrt{-12} = 2i\sqrt{3} \)
