Complex Numbers and Roots Warm Up Lesson Presentation
Complex Numbers and Roots Warm Up Lesson Presentation Lesson Examples Holt Mc. Dougal Algebra 2 Algebra 22 Holt Mc. Dougal
Complex Numbers and Roots Warm up: Holt Mc. Dougal Algebra 2
Complex Numbers and Roots Holt Mc. Dougal Algebra 2
Complex Numbers and Roots Warm Up --- What do we do here? Simplify each expression. 1. Holt Mc. Dougal Algebra 2 2. 3.
Complex Numbers and Roots What about? • This is where the concept of imaginary numbers come in…. . Square root of a negative number? Now we can do these with imaginary numbers Holt Mc. Dougal Algebra 2
Complex Numbers and Roots Lesson Objectives Define and use imaginary and complex numbers. Holt Mc. Dougal Algebra 2
Complex Numbers and Roots Vocabulary Holt Mc. Dougal Algebra 2
Complex Numbers and Roots Holt Mc. Dougal Algebra 2
Complex Numbers and Roots Like ex 1 A: Simplifying Square Roots of Negative Numbers Rewriting them as imaginary numbers using i Express the number in terms of i. Factor out – 1. Product Property. Simplify. Multiply. Express in terms of i. Holt Mc. Dougal Algebra 2
Complex Numbers and Roots Like ex 1 B: Express the number in terms of i. Factor out – 1. Product Property. Simplify. Express in terms of i. Note: Rewrite i in front of radical sign so it does not look like its inside the radical Holt Mc. Dougal Algebra 2
Complex Numbers and Roots Check It Out! Ex. 1 a Express the number in terms of i. Factor out – 1. Product Property. Simplify. Express in terms of i. Holt Mc. Dougal Algebra 2
Complex Numbers and Roots Check It Out! ex 1 b Express the number in terms of i. Factor out – 1. Product Property. Simplify. Multiply. Express in terms of i. Holt Mc. Dougal Algebra 2
Complex Numbers and Roots Check It Out! Example 1 c Express the number in terms of i. Factor out – 1. Product Property. Simplify. Multiply. Express in terms of i. Holt Mc. Dougal Algebra 2
Complex Numbers and Roots Example 2 A: Solving a Quadratic Equation with Imaginary Solutions Solve the equation. Take square roots. Express in terms of i. Check x 2 = – 144 (12 i)2 – 144 i 2 – 144(– 1) – 144 Holt Mc. Dougal Algebra 2 x 2 = (– 12 i)2 144 i 2 144(– 1) – 144
Complex Numbers and Roots Example 2 B: Solving a Quadratic Equation with Imaginary Solutions Solve the equation. 5 x 2 + 90 = 0 Add – 90 to both sides. Divide both sides by 5. Take square roots. Express in terms of i. Check 5 x 2 + 90 = 0 0 5(18)i 2 +90 0 90(– 1) +90 0 Holt Mc. Dougal Algebra 2
Complex Numbers and Roots Check It Out! Example 2 a Solve the equation. x 2 = – 36 Take square roots. Express in terms of i. Check x 2 = – 36 (6 i)2 36 i 2 36(– 1) Holt Mc. Dougal Algebra 2 – 36 x 2 = – 36 (– 6 i)2 – 36 36 i 2 – 36 36(– 1) – 36
Complex Numbers and Roots Check It Out! Example 2 b Solve the equation. x 2 + 48 = 0 x 2 = – 48 Add – 48 to both sides. Take square roots. Express in terms of i. Check x 2 + 48 = + 48 (48)i 2 + 48 48(– 1) + 48 Holt Mc. Dougal Algebra 2 0 0
Complex Numbers and Roots Check It Out! Example 2 c Solve the equation. 9 x 2 + 25 = 0 9 x 2 = – 25 Add – 25 to both sides. Divide both sides by 9. Take square roots. Express in terms of i. Holt Mc. Dougal Algebra 2
Complex Numbers and Roots 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. Holt Mc. Dougal Algebra 2
Complex Numbers and Roots Real numbers are complex numbers where b = 0. ex: 6 can be written as 6 +0 i Imaginary numbers are complex numbers where a = 0 and b ≠ 0. These are sometimes called pure imaginary numbers. Ex: 5 i, -i Holt Mc. Dougal Algebra 2
Complex Numbers and Roots Stop here Holt Mc. Dougal Algebra 2
Complex Numbers and Roots Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. Holt Mc. Dougal Algebra 2
Complex Numbers and Roots Example 3: Equating Two Complex Numbers Find the values of x and y that make the equation 4 x + 10 i = 2 – (4 y)i true. Real parts 4 x + 10 i = 2 – (4 y)i Imaginary parts 4 x = 2 Equate the real parts. Solve for x. Holt Mc. Dougal Algebra 2 10 = – 4 y Equate the imaginary parts. Solve for y.
Complex Numbers and Roots Check It Out! Example 3 a Find the values of x and y that make each equation true. 2 x – 6 i = – 8 + (20 y)i Real parts 2 x – 6 i = – 8 + (20 y)i Imaginary parts 2 x = – 8 Equate the real parts. x = – 4 Solve for x. Holt Mc. Dougal Algebra 2 – 6 = 20 y Equate the imaginary parts. Solve for y.
Complex Numbers and Roots Check It Out! Example 3 b Find the values of x and y that make each equation true. – 8 + (6 y)i = 5 x – i Real parts – 8 + (6 y)i = 5 x – i Imaginary parts – 8 = 5 x Equate the real parts. Solve for x. Holt Mc. Dougal Algebra 2 Equate the imaginary parts. Solve for y.
Complex Numbers and Roots Finding a complex conjugate The real part is the same value and the imaginary part is the opposite sign Find each complex conjugate. B. 6 i A. 8 + 5 i 8 – 5 i Write as a + bi. Find a – bi. Holt Mc. Dougal Algebra 2 0 + 6 i 0 – 6 i Write as a + bi. Find a – bi. Simplify.
Complex Numbers and Roots Check It Out! Example 5 Find each complex conjugate. B. A. 9 – i Write as a + bi. 9+i Find a – bi. C. – 8 i 0 + 8 i 8 i Holt Mc. Dougal Algebra 2 Simplify.
Complex Numbers and Roots Lesson examples to try and bring to class 1. Express in terms of i. 2. Find the values of x and y that make the equation 3 x +8 i = 12 – (12 y)i true. 3. Find the complex conjugate of Holt Mc. Dougal Algebra 2 -8
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