section 7.5 complex numbers and solving quadratic equations with complex solutions
TRANSCRIPT
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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 1i . So:
2i 3i 4i
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.
1. 2. 3.
4. 5. 6.
5i 17i 22i
60i 107i 801i
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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 ______.
1i
a bi 2 7i
Complex Numbers
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Simplify each expression and write the result in the standard a bi form.
7. 36 8. 6
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Simplify each expression and write the result in the standard a bi form.
9. 6 6 10. 81
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Simplify each expression and write the result in the standard a bi form.
11. 81 81 12. 81 81
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Simplify each expression and write the result in the standard a bi form.
13. 100
25
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Simplify each expression and write the result in the standard a bi form.
14. 6 5 6
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Simplify each expression and write the result in the standard a bi form.
15. 36 25 49
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Objective 2: Add, subtract, multiply, and divide complex numbers.
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Addition of Binomials
5 7 3 4 5 3 7 4
8 3
x y x y x x y y
x y
5 7 3 4 5 3 7 4
8 3
i i i i
i
Addition of Complex Numbers
The arithmetic of complex numbers is very similar to the arithmetic of binomials
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16. 3 7 9 8i i
Simplify each expression and write the result in the standard form. a bi
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17.
Simplify each expression and write the result in the standard form.
4 7 3 2i i
a bi
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18. 6 7 3i
Simplify each expression and write the result in the standard a bi form.
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19.
Simplify each expression and write the result in the standard a bi form.
3 10i i
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20.
Simplify each expression and write the result in the standard a bi form.
2 7 9i i
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21.
Simplify each expression and write the result in the standard a bi form.
4 3 5i i
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22.
Simplify each expression and write the result in the standard a bi form.
7 6 3 8i i
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23.
Simplify each expression and write the result in the standard a bi form.
22 5i
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Complex Conjugates: The conjugate of a bi is .a bi
Write the conjugate of each expression. Then multiply the expression by its conjugate.
Expression Conjugate
Product
24. 5 2i
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Complex Conjugates: The conjugate of a bi is .a bi
Write the conjugate of each expression. Then multiply the expression by its conjugate.
Expression Conjugate
Product
25. 8i
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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 a bi form.
Example 3 1 2i i
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26.
Perform the indicated operations and express the result in a bi form.
50 1 3i
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27.
Perform the indicated operations and express the result in a bi form.
2
3 4i
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28. 5 3
2 7
i
i
Perform the indicated operations and express the result in a bi form.
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29. 6 8
1
i
i
Perform the indicated operations and express the result in a bi form.
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30. Determine whether or not is a solution of the equation 2 6 13 0.x x
2 5x i
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31. Determine whether or not is a solution of the equation
3 2x i 2 6 13 0.x x
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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 2x k are x kand x k
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Solve each quadratic equation by extraction of roots.
32. 2 16x
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Solve each quadratic equation by extraction of roots.
33. 2 7x
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Solve each quadratic equation by extraction of roots.
34. 21 4x
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Solve each quadratic equation by extraction of roots.
35. 23 49x
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Solve each quadratic equation by extraction of roots.
36. 22 3 49x
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Solve each quadratic equation by extraction of roots.
37. 23 2 10x
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Recall solving quadratic equations by the Quadratic Formula from Section 7.3: The solutions of the quadratic equation
2 0ax bx c with real coefficients a, b, and c, when 0,a
are 2 4
.2
b b acx
a
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Use the quadratic formula to solve each quadratic equation.
38. 2 2 6 0x x
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39. 23 2 4x x
Use the quadratic formula to solve each quadratic equation.
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40. 2 10 25x x
Use the quadratic formula to solve each quadratic equation.
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41. 3 4 5x x
Use the quadratic formula to solve each quadratic equation.
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42.
Use the quadratic formula to solve each quadratic equation.
22 2 4 0x x x (Hint: Use the zero factor principle.)
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Construct a quadratic equation in x that has the given solutions.
43. and 5x i 5x i
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Construct a quadratic equation in x that has the given solutions.
44. and 3x i 3x i
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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) 2 1 0x
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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) 2 2 1 0x x
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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) 2 1 0x