Presentation on theme: "Complex Numbers and Roots"— Presentation transcript:
1 Complex Numbers and Roots 5-5Complex Numbers and RootsWarm UpLesson PresentationLesson QuizHolt Algebra 2
2 Simplify each expression. Warm UpSimplify each expression.2.1.3.Find the zeros of each function.4.f(x) = x2 – 18x + 165.f(x) = x2 + 8x – 24
3 Objectives Define and use imaginary and complex numbers. Solve quadratic equations with complex roots.
4 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.
6 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.
7 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.
8 Express the number in terms of i. Check It Out! Example 1cExpress the number in terms of i.Factor out –1.Product Property.Simplify.Multiply.Express in terms of i.
9 Example 2B: Solving a Quadratic Equation with Imaginary Solutions Solve the equation.5x = 0Add –90 to both sides.Divide both sides by 5.Take square roots.Express in terms of i.5x = 05(18)i 2 +9090(–1) +90Check
10 Check It Out! Example 2b Solve the equation. x2 + 48 = 0 x2 = –48 Add –48 to both sides.Take square roots.Express in terms of i.Checkx = 0+ 48(48)i48(–1) + 48
11 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.
12 Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.
13 Example 3: Equating Two Complex Numbers Find the values of x and y that make the equation 4x + 10i = 2 – (4y)i true .Real parts4x + 10i = 2 – (4y)iImaginary partsEquate the imaginary parts.Equate the real parts.10 = –4y4x = 2Solve for y.Solve for x.
14 Find the values of x and y that make each equation true. Check It Out! Example 3aFind the values of x and y that make each equation true.2x – 6i = –8 + (20y)iReal parts2x – 6i = –8 + (20y)iImaginary partsEquate the real parts.Equate the imaginary parts.2x = –8–6 = 20yx = –4Solve for y.Solve for x.
15 Example 4A: Finding Complex Zeros of Quadratic Functions Find the zeros of the function.f(x) = x2 + 10x + 26x2 + 10x + 26 = 0Set equal to 0.x2 + 10x = –26 +Rewrite.x2 + 10x + 25 = –Add to both sides.(x + 5)2 = –1Factor.Take square roots.Simplify.
16 Example 4B: Finding Complex Zeros of Quadratic Functions Find the zeros of the function.g(x) = x2 + 4x + 12x2 + 4x + 12 = 0Set equal to 0.x2 + 4x = –12 +Rewrite.x2 + 4x + 4 = –12 + 4Add to both sides.(x + 2)2 = –8Factor.Take square roots.Simplify.
17 Find the zeros of the function. Check It Out! Example 4bFind the zeros of the function.g(x) = x2 – 8x + 18x2 – 8x + 18 = 0Set equal to 0.x2 – 8x = –18 +Rewrite.x2 – 8x + 16 = –Add to both sides.Factor.Take square roots.Simplify.
18 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
19 Example 5: Finding Complex Zeros of Quadratic Functions Find each complex conjugate.B. 6iA i0 + 6i8 + 5iWrite as a + bi.Write as a + bi.0 – 6iFind a – bi.8 – 5iFind a – bi.–6iSimplify.
20 Find each complex conjugate. Check It Out! Example 5Find each complex conjugate.A. 9 – iB.9 + (–i)Write as a + bi.Write as a + bi.9 – (–i)Find a – bi.Find a – bi.9 + iSimplify.C. –8i0 + (–8)iWrite as a + bi.0 – (–8)iFind a – bi.8iSimplify.
21 Lesson Quiz1. Express in terms of i.Solve each equation.3. x2 + 8x +20 = 02. 3x = 04. Find the values of x and y that make the equation 3x +8i = 12 – (12y)i true.5. Find the complex conjugate of