1. Complex Numbers 3.3 Perform arithmetic operations on complex numbers
Solve quadratic equations having complex solutions
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2. Properties of the Imaginary Unit i
Defining the number i allows us to say that the solutions to the equation x2 + 1 = 0 are i and –i.
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3. Complex Numbers
A complex number can be written in standard form as a + bi where a and b are real numbers. The real part is a and the imaginary part is b. Every real number a is also a complex number because it can be written as a + 0i.
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4. Imaginary Numbers
A complex number a + bi with b ≠ 0 is an imaginary number. A complex number a + bi with a = 0 and b ≠ 0 is sometimes called a pure imaginary number. Examples of pure imaginary numbers include 3i and –i.
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5. The Expression If a > 0, then
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6. Example 2 Simplify each expression. Solution
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7. Example 3
Write each expression in standard form. Support your results using a calculator.
a) (3 + 4i) + (5  i) b) (7i)  (6  5i)
c) (3 + 2i)2 d)
Solution
a) (3 + 4i) + (5  i) =  i  i = 2 + 3i
b) (7i)  (6  5i) = 6  7i + 5i = 6  2i
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8. Solution continued c) (3 + 2i)2 = (3 + 2i)(3 + 2i)
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9. Quadratic Equations with Complex Solutions
We can use the quadratic formula to solve quadratic equations if the discriminant is negative.
There are no real solutions, and the graph does not intersect the x-axis.
The solutions can be expressed as imaginary numbers.
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10. Example 4a Solve the quadratic equation
Support your answer graphically.
Solution
Rewrite the equation:
a = 1/2, b = –5, c = 17
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11. Example 4a Solution continued
The graphs do not intersect, so no real solutions, but two complex solutions that are imaginary.
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12. Example 4b
Solve the quadratic equation x2 + 3x + 5 = 0. Support your answer graphically.
Solution
a = 1, b = 3, c = 5
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13. Example 4b Solution continued
The graph does not intersect the x-axis, so no real solutions, but two complex solutions that are imaginary.
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14. Example 4c Solve the quadratic equation –2x2 = 3.
Support your answer graphically.
Solution
Apply the square root property.
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15. Example 4c Solution continued
The graphs do not intersect, so no real solutions, but two complex solutions that are imaginary.
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