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5-5 Complex Numbers and Roots Solve the equation: x2 + 16 = 0 x2 = -16 Isolate the variable 𝑥2 = Square root each side −16 There is not a real solution to the square root of a negative number. Instead, a new set of numbers are required. Imaginary numbers are defined as the square root of negative numbers. The imaginary unit: i = Holt Algebra 2 −1 5-5 Complex Numbers and Roots X-intercepts are defined as the solutions to quadratic functions. Parabolas that do not have x-intercepts have imaginary solutions. Holt Algebra 2 5-5 Complex Numbers and Roots For any positive real number b, 2 2 b b 1 bi where i is the imaginary unit and bi is called the pure imaginary number. Holt Algebra 2 5-5 i0 i1 i2 i3 i4 i5 i6 i7 Complex Numbers and Roots Cycle of i =1 =i 2 = −1 =-1 3 = −1 = −1 =1 =i = -1 =-i Holt Algebra 2 2 −1 = -i 5-5 Complex Numbers and Roots 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. Holt Algebra 2 5-5 Complex Numbers and Roots Example 1B: Simplifying Square Roots of Negative Numbers Express the number in terms of i. Factor out –1. Product Property. Simplify. 4 6i 4i 6 Holt Algebra 2 Express in terms of i. 5-5 Complex Numbers and Roots Check It Out! Example 1a Express the number in terms of i. Factor out –1. Product Property. Product Property. Simplify. Express in terms of i. Holt Algebra 2 5-5 Complex Numbers and Roots Check It Out! Example 1b Express the number in terms of i. Factor out –1. Product Property. Simplify. Multiply. Express in terms of i. Holt Algebra 2 5-5 Complex Numbers and Roots Example 2A: Solving a Quadratic Equation with Imaginary Solutions Solve the equation. Take square roots. Express in terms of i. Check x2 = –144 (12i)2 –144 144i 2 –144 144(–1) –144 Holt Algebra 2 x2 = (–12i)2 144i 2 144(–1) –144 –144 –144 –144 5-5 Complex Numbers and Roots Example 2B: Solving a Quadratic Equation with Imaginary Solutions Solve the equation. 5x2 + 90 = 0 Add –90 to both sides. Divide both sides by 5. Take square roots. Express in terms of i. Check 5x2 + 90 = 0 0 5(18)i 2 +90 0 90(–1) +90 0 Holt Algebra 2 5-5 Complex Numbers and Roots Complex numbers are numbers written in the form: a + bi Every complex number has a real part a and an imaginary part b. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. Holt Algebra 2 5-5 Complex Numbers and Roots Find the values of x and y that make the equation 4x + 10i = 2 – (4y)i true . Real parts 4x + 10i = 2 – (4y)i 4x = 2 Imaginary parts Equate the 10 = –4y Equate the imaginary parts. real parts. Solve for x. Holt Algebra 2 Solve for y. 5-5 Complex Numbers and Roots Example 4A: Finding Complex Zeros of Quadratic Functions Find the zeros of the function. f(x) = x2 + 10x + 26 x2 + 10x + 26 = 0 Set equal to 0. x2 + 10x + Rewrite. = –26 + x2 + 10x + 25 = –26 + 25 (x + 5)2 = –1 Add to both sides. Factor. Take square roots. Simplify. Holt Algebra 2 5-5 Complex Numbers and Roots Example 4B: Finding Complex Zeros of Quadratic Functions Find the zeros of the function. g(x) = x2 + 4x + 12 x2 + 4x + 12 = 0 Set equal to 0. x2 + 4x + Rewrite. = –12 + x2 + 4x + 4 = –12 + 4 (x + 2)2 = –8 Add to both sides. Factor. Take square roots. Simplify. Holt Algebra 2 5-5 Complex Numbers and Roots Check It Out! Example 4a Find the zeros of the function. f(x) = x2 + 4x + 13 x2 + 4x + 13 = 0 Set equal to 0. x2 + 4x + Rewrite. = –13 + x2 + 4x + 4 = –13 + 4 (x + 2)2 = –9 Add to both sides. Factor. Take square roots. x = –2 ± 3i Holt Algebra 2 Simplify. 5-5 Complex Numbers and Roots HW pg. 353 #’s 18-25, 36, 46-51 Holt Algebra 2