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Mastering Complex Numbers in Quadratic Equations

Learn how to add, subtract, multiply, and divide complex numbers and solve quadratic equations in the complex number system. Understand the concept of imaginary unit and operations with complex numbers. Discover the formula to find solutions to quadratic equations in the real or complex number system.

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Mastering Complex Numbers in Quadratic Equations

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  1. Sullivan Algebra and Trigonometry: Section 1.3Quadratic Equations in the Complex Number System • Objectives • Add, Subtract, Multiply, and Divide Complex Numbers • Solve Quadratic Equations in the Complex Number System

  2. The equation x2 = - 1 has no real number solution. To remedy this situation, we define a new number that solves this equation, called the imaginary unit, which is not a real number. The solution to x2 = - 1 is the imaginary unit I where i2 = - 1, or

  3. Complex numbers are numbers of the form a + bi, where a and b are real numbers. The real number a is called the real part of the number a + bi; the real number b is called the imaginary part of a + bi. Equality of Complex Numbers a + bi = c + di if and only if a = c and b = d In other words, complex numbers are equal if and only if there real and imaginary parts are equal.

  4. Addition with Complex Numbers (a + bi) + (c + di) = (a + c) + (b + d)i Example: (2 + 4i) + (-1 + 6i) = (2 - 1) + (4 + 6)i = 1 + 10i

  5. Subtraction with Complex Numbers (a + bi) - (c + di) = (a - c) + (b - d)i Example (3 + i) - (1 - 2i) = (3 - 1) + (1 - (-2))i = 2 + 3i

  6. Multiplication with Complex Numbers Example: Multiply using the distributive property

  7. If z = a + bi is a complex number, then its conjugate, denoted by , is defined as Theorem The product of a complex number and its conjugate is a nonnegative real number. Thus, if z = a + bi, then

  8. Division with Complex Numbers To divide by a complex number, multiply the dividend (numerator) and divisor (denominator) by the conjugate of the divisor. Example:

  9. In the complex number system, the solutions of the quadratic equation where a, b, and c are real numbers and a 0, are given by the formula Since we now have a way of evaluating the square root of a negative number, there are now no restrictions placed on the quadratic formula.

  10. Find all solutions to the equation real or complex.

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