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Algebra 2 Unit 5 Imaginary and Complex Numbers Content to be Learned: Definition of the imaginary number, i Simplifying negative radical expressions Quadratic formula ~ Imaginary numbers Complex numbers Operations on complex numbers The Imaginary Number, i Recall that if you graph a quadratic function, you get one of three results: Two Real Zeros One Real Zero No Real Zeros (means that the zeros are imaginary) We have looked at (and solved) quadratic equations that either have 2 solutions (and cross the x-axis twice) or 1 solution (cross the x-axis once). Now, what would the quadratic formula look like for a graph that has no real solutions, meaning that the graph never crosses the x-axis and the solutions are imaginary? First, we must understand what the imaginary number is. By definition, i= The way it is used is as follows: Recall: 25 5 1 and i2 = -1 Now, 25 25 1 25 1 5i Try simplifying the following: 100 36 24 50 What would 3i2 simplify to? __________________ How about (3i)2_________________ Now that you can simplify radicals of negative numbers, you can solve quadratic equations whose graphs do not cross the x-axis, meaning that the roots (zeros) are imaginary)!! Example: Use the quadratic formula to solve 3x2 + x + 2 = 0. Practice Directions: Simplify each expression. 9 2) 24 300 4) 75 1) 3) 63 6) 225 7) 112 8) 99 5) Practice Complex Numbers and Operations A complex number is ______________________________________________ ________________________________________________________________ We can perform operations on complex numbers very much like how we perform operations on algebraic expressions. Remember: i2 = -1 and 1 = i Adding and Subtracting Complex Numbers Example 1: Add (7 + 5i) + (8 - 3i) Answer: 15 + 2i Example 2: Add (-3 + 4i ) + (7i + 5) Answer: Example 3: Add (4 – 3i ) + (-2 – i) Answer: Example 4: Subtract: (5 + 6i ) – (2 + 3i ) Answer: 3 + 3i Example 5: Subtract: (-4i – 3) – (5 + 2i ) Answer: Multiplying Complex Numbers- Think FOIL Example 1: Multiply (3 + 2i )(1 + 4i ) Solution: 3 + 12i + 2i + 8i 2 3 + 14i + 8(-1) 3 + 14i – 8 -5 + 14i Example 3: Multiply (3 + 7i )(4i – 8) Example 4: i(2 + i)(3 – 4i) + 2i Example 2: Multiply (-2 – 3i )(-5 + i ) Practice Division of Complex numbers In order to divide complex numbers, we need to know about conjugates. The conjugate of a + bi is a – bi The conjugate of a – bi is a + bi Examples: What is the conjugate of 2 -3i ?_________________________ What is the conjugate of 2 + 3i ?_________________________ What is the conjugate of -2 -3i ?_________________________ What is the conjugate of -2 + 3i ?_________________________ Example: Simplify (-3 – 4i) (2 + 3i) Practice