# Complex Numbers and Solving Quadratic Equations with Complex Solutions

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Complex Numbers and Solving Quadratic Equations with Complex Solutions
Section 7.5 Complex Numbers and Solving Quadratic Equations with Complex Solutions

Objective 1: Express complex numbers in standard form.
The Imaginary Number i The imaginary number i is defined as . So:

Notice that any power of i can be simplified to i, −1, −i, or 1
Notice that any power of i can be simplified to i, −1, −i, or 1. Simplify each power of i. 1. 2. 3. 4. 5. 6.

Complex Numbers If a and b are real numbers and , then: Algebraic Form Numerical Example is a complex number with a ____________ term a and an ____________ term bi. has a real term ______ and an imaginary term ______.

Simplify each expression and write the result in the standard
form. 7. 8.

Simplify each expression and write the result in the standard
form. 9. 10.

Simplify each expression and write the result in the standard
form. 11. 12.

Simplify each expression and write the result in the standard
form. 13.

Simplify each expression and write the result in the standard
form. 14.

Simplify each expression and write the result in the standard
form. 15.

Objective 2: Add, subtract, multiply, and divide complex numbers.

The arithmetic of complex numbers is very similar to the arithmetic of binomials

Simplify each expression and write the result in the standard
form. 16.

Simplify each expression and write the result in the standard
form. 17.

Simplify each expression and write the result in the standard
form. 18.

Simplify each expression and write the result in the standard
form. 19.

Simplify each expression and write the result in the standard
form. 20.

Simplify each expression and write the result in the standard
form. 21.

Simplify each expression and write the result in the standard
form. 22.

Simplify each expression and write the result in the standard
form. 23.

Complex Conjugates: The conjugate of
is Write the conjugate of each expression. Then multiply the expression by its conjugate. Expression Conjugate 24. Product

Complex Conjugates: The conjugate of
is Write the conjugate of each expression. Then multiply the expression by its conjugate. Expression Conjugate 25. Product

Dividing Complex Numbers - The fact that the product of a complex number and its conjugate is always a real number plays a key role in the division of complex numbers as outlined in the following box.

Steps for Dividing Complex Numbers
Example Step 1. Write the division problem as a fraction. Step 2. Multiply both the numerator and the denominator by the conjugate of the denominator. Step 3. Simplify the result, and express it in standard form.

Perform the indicated operations and express the result in
26.

Perform the indicated operations and express the result in
27.

Perform the indicated operations and express the result in
28.

Perform the indicated operations and express the result in
29.

30. Determine whether or not is a solution of the
equation

31. Determine whether or not is a solution of the
equation

Objective 3: Solve a quadratic equation with imaginary solutions
Recall solving quadratic equations by extraction of roots from Section 7.1: The solutions of are and

Solve each quadratic equation by extraction of roots.
32.

Solve each quadratic equation by extraction of roots.
33.

Solve each quadratic equation by extraction of roots.
34.

Solve each quadratic equation by extraction of roots.
35.

Solve each quadratic equation by extraction of roots.
36.

Solve each quadratic equation by extraction of roots.
37.

Recall solving quadratic equations by the Quadratic Formula from Section 7.3: The solutions of the quadratic equation with real coefficients a, b, and c, when are

38.

39.

40.

41.

42. (Hint: Use the zero factor principle.)

Construct a quadratic equation in x that has the given solutions.
43. and

Construct a quadratic equation in x that has the given solutions.
44. and

45. Determine the discriminant of each of these quadratic
equations and then determine the solution of each equation. Equation Discriminant Solution (a)

45. Determine the discriminant of each of these quadratic
equations and then determine the solution of each equation. Equation Discriminant Solution (b)

45. Determine the discriminant of each of these quadratic
equations and then determine the solution of each equation. Equation Discriminant Solution (c)