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Section 7.5 Complex Numbers and Solving Quadratic Equations with Complex Solutions

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

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Notice that any power of i can be simplified to i, 1, i, or 1. Simplify each power of i

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

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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

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Simplify each expression and write the result in the standard form

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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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Addition of Binomials Addition of Complex Numbers The arithmetic of complex numbers is very similar to the arithmetic of binomials

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

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

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

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

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

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

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

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

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Complex Conjugates: The conjugate ofis Write the conjugate of each expression. Then multiply the expression by its conjugate. ExpressionConjugate Product 24.

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Complex Conjugates: The conjugate ofis Write the conjugate of each expression. Then multiply the expression by its conjugate. ExpressionConjugate Product 25.

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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 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. Example

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26. Perform the indicated operations and express the result in form.

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27. Perform the indicated operations and express the result in form.

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28. Perform the indicated operations and express the result in form.

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29. Perform the indicated operations and express the result in form.

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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. 32.

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Solve each quadratic equation by extraction of roots. 33.

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Solve each quadratic equation by extraction of roots. 34.

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Solve each quadratic equation by extraction of roots. 35.

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Solve each quadratic equation by extraction of roots. 36.

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Solve each quadratic equation by extraction of roots. 37.

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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. 38.

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

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

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

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42. Use the quadratic formula to solve each quadratic equation. (Hint: Use the zero factor principle.)

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

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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. EquationDiscriminantSolution (a)

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

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

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