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**Simplify each expression.**

Warm Up Simplify each expression. 2. 1. 3. Find the zeros of each function. 4. f(x) = x2 – 18x + 16 5. f(x) = x2 + 8x – 24

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Ch. 4.6 : I Can define and use imaginary and complex numbers and solve quadratic equations with complex roots Do Now: Find the zeros of each function. 1. f(x) = x2 – 18x + 16 2. f(x) = x2 + 8x – 24 Success Criteria: Today’s Agenda Check HW Notes Assignment I can use complex numbers I can apply the conjugates

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Ch. 4.6 : I Can define and use imaginary and complex numbers and solve quadratic equations with complex roots Do Now: Find the zeros of each function. 2. x2 + 8x +20 = 0 1. 3x = 0 Success Criteria: Today’s Agenda Notes Assignment I can use complex numbers I can apply the conjugates

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You can see in the graph of f(x) = x2 + 1 below that f has no real zeros. If you solve the corresponding equation 0 = x2 + 1, you find that x = ,which has no real solutions. However, you can find solutions if you define the square root of negative numbers, which is why imaginary numbers were invented. The imaginary unit i is defined as You can use the imaginary unit to write the square root of any negative number.

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**Example 1A: Simplifying Square Roots of Negative Numbers**

Express the number in terms of i. Factor out –1. Product Property. Simplify. Multiply. Express in terms of i.

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**Example 1B: Simplifying Square Roots of Negative Numbers**

Express the number in terms of i. Factor out –1. Product Property. Simplify. Express in terms of i.

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Check It Out! Example 1a Express the number in terms of i.

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A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i = The set of real numbers is a subset of the set of complex numbers C. Every complex number has a real part a and an imaginary part b.

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**Example 3A: Adding and Subtracting Complex Numbers**

Add or subtract. Write the result in the form a + bi. (4 + 2i) + (–6 – 7i) (4 – 6) + (2i – 7i) Add real parts and imaginary parts. –2 – 5i

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**Example 3B: Adding and Subtracting Complex Numbers**

Add or subtract. Write the result in the form a + bi. (5 –2i) – (–2 –3i) (5 – 2i) i Distribute. (5 + 2) + (–2i + 3i) Add real parts and imaginary parts. 7 + i

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Check It Out! Example 3a Add or subtract. Write the result in the form a + bi. (–3 + 5i) + (–6i) (–3) + (5i – 6i) Add real parts and imaginary parts. –3 – i

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Check It Out! Example 3b Add or subtract. Write the result in the form a + bi. 2i – (3 + 5i) (2i) – 3 – 5i Distribute. (–3) + (2i – 5i) Add real parts and imaginary parts. –3 – 3i

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**Example 5A: Multiplying Complex Numbers**

Multiply. Write the result in the form a + bi. –2i(2 – 4i) –4i + 8i2 Distribute. –4i + 8(–1) Use i2 = –1. –8 – 4i Write in a + bi form.

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**Example 5B: Multiplying Complex Numbers**

Multiply. Write the result in the form a + bi. (3 + 6i)(4 – i) i – 3i – 6i2 Multiply. i – 6(–1) Use i2 = –1. i Write in a + bi form.

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**Example 5C: Multiplying Complex Numbers**

Multiply. Write the result in the form a + bi. (2 + 9i)(2 – 9i) 4 – 18i + 18i – 81i2 Multiply. 4 – 81(–1) Use i2 = –1. 85 Write in a + bi form.

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**The solutions and are related**

The solutions and are related. These solutions are a complex conjugate pair. Their real parts are equal and their imaginary parts are opposites. The complex conjugate of any complex number a + bi is the complex number a – bi. If a quadratic equation with real coefficients has nonreal roots, those roots are complex conjugates. When given one complex root, you can always find the other by finding its conjugate. Helpful Hint

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**Example 5: Finding Complex Zeros of Quadratic Functions**

Find each complex conjugate. B. 6i A i 0 + 6i 8 + 5i Write as a + bi. Write as a + bi. 0 – 6i Find a – bi. 8 – 5i Find a – bi. –6i Simplify. C. 9 – i D. 9 + (–i) Write as a + bi. Write as a + bi. 9 – (–i) Find a – bi. Find a – bi. 9 + i Simplify.

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**Use complex conjugates to simplify**

Multiply by the conjugate. Distribute. Use i2 = –1. Simplify.

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**Use complex conjugates to simplify**

Multiply by the conjugate. Distribute. Use i2 = –1. Simplify.

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Lesson Quiz: Part II Perform the indicated operation. Write the result in the form a + bi. 1. (2 + 4i) + (–6 – 4i) –4 2. (5 – i) – (8 – 2i) –3 + i 3. (2 + 5i)(3 – 2i) 4. 3 + i i 5. Simplify i31. –i

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Assignment #47 p #9-12, 18 – 26, 27, 29, 31

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Ch. 4.6 : I Can define and use imaginary and complex numbers and solve quadratic equations with complex roots Do Now: The base of a triangle is 3 cm longer than its altitude. The area of the triangle is 35 cm2. Find the altitude. Hint: draw the triangle and use formula. Success Criteria: Today’s Agenda Notes Assignment I can use complex numbers I can apply the conjugates

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**Example 2A: Solving a Quadratic Equation with Imaginary Solutions**

Solve the equation. Take square roots. Express in terms of i. Check x2 = –144 –144 (12i)2 144i 2 144(–1) x2 = –144 –144 144(–1) 144i 2 (–12i)2

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**Example 2B: Solving a Quadratic Equation with Imaginary Solutions**

Solve the equation. 5x = 0 Add –90 to both sides. Divide both sides by 5. Take square roots. Express in terms of i. 5x = 0 5(18)i 2 +90 90(–1) +90 Check

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Check It Out! Example 2a Solve the equation. x2 = –36 x = 0 x2 = –48 9x = 0 9x2 = –25

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**Example 4A: Finding Complex Zeros of Quadratic Functions**

Find the zeros of the function. f(x) = x2 + 10x + 26 g(x) = x2 + 4x + 12 x2 + 10x + 26 = 0 x2 + 4x + 12 = 0 x2 + 10x = –26 + x2 + 4x = –12 + x2 + 10x + 25 = – x2 + 4x + 4 = –12 + 4 (x + 5)2 = –1 (x + 2)2 = –8

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Check It Out! Example 4a Find the zeros of the function. f(x) = x2 + 4x + 13 g(x) = x2 – 8x + 18 x2 + 4x + 13 = 0 x2 – 8x + 18 = 0 x2 + 4x = –13 + x2 – 8x = –18 + x2 + 4x + 4 = –13 + 4 x2 – 8x + 16 = – (x + 2)2 = –9 x = –2 ± 3i

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Assignment #48 Pg 273 #2,5,6-16, 21-23, 26

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Ch. 4.6 : I Can define and use imaginary and complex numbers and solve quadratic equations with complex roots Do Now: A farmer is fencing a yard in the shape of a trapezoid. She wants the shorter base to be 1 yard greater than the height and the longer base to be 9 yards greater than the height. She wants the area to be 200 square yards. Find the height that will give the desired area, rounded to the nearest hundredth of a yard. Success Criteria: Today’s Agenda Notes Assignment I can use complex numbers I can apply the conjugates

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Assignment #48 Pg 273 #2,5,6-16, 21-23, 26

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Lesson Quiz 1. Express in terms of i. Solve each equation. 3. x2 + 8x +20 = 0 2. 3x = 0 4. Find the complex conjugate of

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