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## 5-5

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**5-5**Complex Numbers and Roots Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2**Warm Up**Simplify each expression. 2. 1. 3. Find the zeros of each function. 4. f(x) = x2– 18x + 16 f(x) = x2+ 8x – 24 5.**Objectives**Define and use imaginary and complex numbers. Solve quadratic equations with complex roots.**Vocabulary**imaginary unit imaginary number complex number real part imaginary part complex conjugate**However, you can find solutions if you define the square**root of negative numbers, which is why imaginary numbers were invented. The imaginary uniti is defined as . You can use the imaginary unit to write the square root of any negative number. 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.**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.**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.**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.**Check It Out! Example 1b**Express the number in terms of i. Factor out –1. Product Property. Simplify. Multiply. Express in terms of i.**Check It Out! Example 1c**Express the number in terms of i. Factor out –1. Product Property. Simplify. Multiply. Express in terms of i.**x2 = –144**x2 = –144 (–12i)2 (12i)2 –144 –144 144i 2 –144 144i 2 –144 –144 144(–1) –144 144(–1) Example 2A: Solving a Quadratic Equation with Imaginary Solutions Solve the equation. Take square roots. Express in terms of i. Check**5x2 + 90 = 0**0 5(18)i 2 +90 0 90(–1) +90 0 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**x2 = –36**x2 = –36 (6i)2 (–6i)2 –36 –36 36i 2 –36 36i 2 –36 –36 –36 36(–1) 36(–1) Check It Out! Example 2a Solve the equation. x2 = –36 Take square roots. Express in terms of i. Check**+ 48**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. Check x2 + 48 = 0 0 (48)i 2+ 48 0 48(–1)+ 48 0 **Check It Out! Example 2c**Solve the equation. 9x2 + 25 = 0 9x2 = –25 Add –25 to both sides. Divide both sides by 9. Take square roots. Express in terms of i.**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 parta and an imaginary partb.**Real numbers are complex numbers where b = 0. Imaginary**numbers are complex numbers where a = 0 and b ≠ 0. These are sometimes called pureimaginary numbers. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.**Example 3: Equating Two Complex Numbers**Find the values of x and y that make the equation 4x + 10i = 2 – (4y)i true . Real parts 4x + 10i = 2 – (4y)i Imaginary parts Equate the imaginary parts. Equate the real parts. 10 = –4y 4x = 2 Solve for y. Solve for x.**Check It Out! Example 3a**Find the values of x and y that make each equation true. 2x – 6i = –8 + (20y)i Real parts 2x– 6i = –8 + (20y)i Imaginary parts Equate the imaginary parts. Equate the real parts. –6 = 20y 2x = –8 Solve for y. x = –4 Solve for x.**–8 + (6y)i = 5x – i**–8 + (6y)i = 5x – i Check It Out! Example 3b Find the values of x and y that make each equation true. Real parts Imaginary parts Equate the imaginary parts. Equate the real parts. –8 = 5x Solve for y. Solve for x.**x2 + 10x + = –26 +**Add to both sides. Example 4A: Finding Complex Zeros of Quadratic Functions Find the zeros of the function. f(x) = x2 + 10x + 26 Set equal to 0. x2 + 10x + 26 = 0 Rewrite. x2 + 10x + 25 = –26 + 25 (x + 5)2 = –1 Factor. Take square roots. Simplify.**x2 + 4x + = –12 +**Add to both sides. 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. Rewrite. x2 + 4x + 4 = –12 + 4 (x + 2)2 = –8 Factor. Take square roots. Simplify.**x2 + 4x + = –13 +**Add to both sides. Check It Out! Example 4a Find the zeros of the function. f(x) = x2 + 4x + 13 x2 + 4x + 13 = 0 Set equal to 0. Rewrite. x2 + 4x + 4 = –13 + 4 (x + 2)2 = –9 Factor. Take square roots. x = –2 ± 3i Simplify.**x2 – 8x + = –18 +**Add to both sides. Check It Out! Example 4b Find the zeros of the function. g(x) = x2 – 8x + 18 x2 – 8x + 18 = 0 Set equal to 0. Rewrite. x2 – 8x + 16 = –18 + 16 Factor. Take square roots. Simplify.**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. Helpful Hint When given one complex root, you can always find the other by finding its conjugate. If a quadratic equation with real coefficients has nonreal roots, those roots are complex conjugates.**Example 5: Finding Complex Zeros of Quadratic Functions**Find each complex conjugate. B. 6i A. 8 + 5i 0 + 6i Write as a + bi. 8 + 5i Write as a + bi. 0 – 6i Find a – bi. 8 – 5i Find a – bi. –6i Simplify.**B.**Check It Out! Example 5 Find each complex conjugate. A. 9 – i 9 + (–i) Write as a + bi. Write as a + bi. 9 – (–i) Find a – bi. Find a – bi. 9 + i Simplify. C. –8i 0 + (–8)i Write as a + bi. 0 – (–8)i Find a – bi. 8i Simplify.**1. Express in terms of i.**Lesson Quiz Solve each equation. 3.x2 + 8x +20 = 0 2. 3x2 + 96 = 0 4. Find the values of x and y that make the equation 3x +8i = 12 – (12y)i true. 5. Find the complex conjugate of