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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

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**Objective 1: Express complex numbers in standard form.**

The Imaginary Number i The imaginary number i is defined as . So:

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**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.

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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 ______.

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**Simplify each expression and write the result in the standard**

form. 7. 8.

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**Simplify each expression and write the result in the standard**

form. 9. 10.

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**Simplify each expression and write the result in the standard**

form. 11. 12.

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**Simplify each expression and write the result in the standard**

form. 13.

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**Simplify each expression and write the result in the standard**

form. 14.

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**Simplify each expression and write the result in the standard**

form. 15.

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

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

Addition of Binomials Addition of Complex Numbers

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**Simplify each expression and write the result in the standard**

form. 16.

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**Simplify each expression and write the result in the standard**

form. 17.

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**Simplify each expression and write the result in the standard**

form. 18.

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**Simplify each expression and write the result in the standard**

form. 19.

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**Simplify each expression and write the result in the standard**

form. 20.

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**Simplify each expression and write the result in the standard**

form. 21.

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**Simplify each expression and write the result in the standard**

form. 22.

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**Simplify each expression and write the result in the standard**

form. 23.

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**Complex Conjugates: The conjugate of**

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

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**Complex Conjugates: The conjugate of**

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

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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.

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**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.

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**Perform the indicated operations and express the result in**

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**Perform the indicated operations and express the result in**

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**Perform the indicated operations and express the result in**

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**Perform the indicated operations and express the result in**

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

equation

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

equation

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**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

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**Solve each quadratic equation by extraction of roots.**

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**Solve each quadratic equation by extraction of roots.**

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**Solve each quadratic equation by extraction of roots.**

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**Solve each quadratic equation by extraction of roots.**

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**Solve each quadratic equation by extraction of roots.**

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**Solve each quadratic equation by extraction of roots.**

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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

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**Use the quadratic formula to solve each quadratic equation.**

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**Use the quadratic formula to solve each quadratic equation.**

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**Use the quadratic formula to solve each quadratic equation.**

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**Use the quadratic formula to solve each quadratic equation.**

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**Use the quadratic formula to solve each quadratic equation.**

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

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

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

44. and

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**45. Determine the discriminant of each of these quadratic**

equations and then determine the solution of each equation. Equation Discriminant Solution (a)

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**45. Determine the discriminant of each of these quadratic**

equations and then determine the solution of each equation. Equation Discriminant Solution (b)

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**45. Determine the discriminant of each of these quadratic**

equations and then determine the solution of each equation. Equation Discriminant Solution (c)

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