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2-5 2-5 Complex Complex Numbers Numbers and and Roots Roots Warm Up Lesson Presentation Lesson Quiz Holt Holt McDougal Algebra 2Algebra Algebra22 Holt McDougal 2-5 Complex Numbers and Roots Warm Up Simplify each expression. 2. 1. 3. Find the zeros of the function by completing the square. 4. f(x) = x2 β 18x + 16 Rewrite the function in vertex form. 5. f(x) = x2 + 8x β 24 Holt McDougal Algebra 2 π π = π+π π β ππ 2-5 Complex Numbers and Roots Objectives Define and use imaginary and complex numbers. Solve quadratic equations with complex roots. Holt McDougal Algebra 2 2-5 Complex Numbers and Roots Vocabulary imaginary unit imaginary number complex number real part imaginary part complex conjugate Holt McDougal Algebra 2 2-5 Complex Numbers and Roots Essential Question β’ How do you find complex zeros of quadratic functions? Holt McDougal Algebra 2 2-5 Complex Numbers and Roots 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. Holt McDougal Algebra 2 2-5 Complex Numbers and Roots Holt McDougal Algebra 2 2-5 Complex Numbers and Roots Example 1A: Simplifying Square Roots of Negative Numbers Express the number in terms of i. Factor out β1. Product Property. Simplify. Multiply. Express in terms of i. Holt McDougal Algebra 2 2-5 Complex Numbers and Roots Example 1B: Simplifying Square Roots of Negative Numbers Express the number in terms of i. Factor out β1. Product Property. Simplify. ο4 6i ο½ ο4i 6 Holt McDougal Algebra 2 Express in terms of i. 2-5 Complex Numbers and Roots Check It Out! Example 1a Express the number in terms of i. Factor out β1. Product Property. Product Property. Simplify. Express in terms of i. Holt McDougal Algebra 2 2-5 Complex Numbers and Roots Check It Out! Example 1b Express the number in terms of i. Factor out β1. Product Property. Simplify. Multiply. Express in terms of i. Holt McDougal Algebra 2 2-5 Complex Numbers and Roots Check It Out! Example 1c Express the number in terms of i. Factor out β1. Product Property. Simplify. Multiply. Express in terms of i. Holt McDougal Algebra 2 2-5 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 = β144 (12i)2 β144 144i 2 β144 144(β1) β144 οΌ Holt McDougal Algebra 2 x2 = (β12i)2 144i 2 144(β1) β144 β144 β144 β144 οΌ 2-5 Complex Numbers and Roots Example 2B: Solving a Quadratic Equation with Imaginary Solutions Solve the equation. 5x2 + 90 = 0 Add β90 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(β1) +90 0 οΌ Holt McDougal Algebra 2 2-5 Complex Numbers and Roots Check It Out! Example 2a Solve the equation. x2 = β36 Take square roots. Express in terms of i. Check x2 = β36 (6i)2 36i 2 36(β1) Holt McDougal Algebra 2 β36 β36 β36 οΌ x2 = β36 (β6i)2 β36 36i 2 β36 36(β1) β36 οΌ 2-5 Complex Numbers and Roots Check It Out! Example 2b Solve the equation. x2 + 48 = 0 x2 = β48 Add β48 to both sides. Take square roots. Express in terms of i. Check x2 + 48 = + 48 (48)i 2 + 48 48(β1) + 48 Holt McDougal Algebra 2 0 0 0 0οΌ 2-5 Complex Numbers and Roots Check It Out! Example 2c Solve the equation. 9x2 + 25 = 0 9x2 = β25 Add β25 to both sides. Divide both sides by 9. Take square roots. Express in terms of i. Holt McDougal Algebra 2 2-5 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 McDougal Algebra 2 2-5 Complex Numbers and Roots Real numbers are complex numbers where b = 0. 4 Examples: 4, β10, , 0.5, and 15. 3 Imaginary numbers are complex numbers where a = 0 and b β 0. These are sometimes called pure imaginary numbers. 2 Examples: π, 4i, β π, π 6. 5 Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. Holt McDougal Algebra 2 2-5 Complex Numbers and Roots Example 3: Equating Two Complex Numbers Find the values of x and y that make the equation 4x + 10i = 2 β (4y)i true . Real parts 4x + 10i = 2 β (4y)i Imaginary parts 4x = 2 Equate the real parts. Solve for x. Holt McDougal Algebra 2 10 = β4y Equate the imaginary parts. Solve for y. 2-5 Complex Numbers and Roots Check It Out! Example 3a Find the values of x and y that make each equation true. 2x β 6i = β8 + (20y)i Real parts 2x β 6i = β8 + (20y)i Imaginary parts 2x = β8 Equate the real parts. x = β4 Solve for x. Holt McDougal Algebra 2 β6 = 20y Equate the imaginary parts. Solve for y. 2-5 Complex Numbers and Roots Check It Out! Example 3b Find the values of x and y that make each equation true. β8 + (6y)i = 5x β i 6 Real parts β8 + (6y)i = 5x β i 6 Imaginary parts β8 = 5x Equate the real parts. Solve for x. Holt McDougal Algebra 2 Equate the imaginary parts. Solve for y. 2-5 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. = β26 + x2 + 10x + 25 = β26 + 25 (x + 5)2 = β1 Add to both sides. Factor. Take square roots. Simplify. Holt McDougal Algebra 2 2-5 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. = β12 + x2 + 4x + 4 = β12 + 4 (x + 2)2 = β8 Add to both sides. Factor. Take square roots. Simplify. Holt McDougal Algebra 2 2-5 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. = β13 + x2 + 4x + 4 = β13 + 4 (x + 2)2 = β9 Add to both sides. Factor. Take square roots. x = β2 ± 3i Holt McDougal Algebra 2 Simplify. 2-5 Complex Numbers and Roots Check It Out! Example 4b Find the zeros of the function. g(x) = x2 β 8x + 18 x2 β 8x + 18 = 0 Set equal to 0. x2 β 8x + Rewrite. = β18 + x2 β 8x + 16 = β18 + 16 Add to both sides. Factor. Take square roots. Simplify. Holt McDougal Algebra 2 2-5 Complex Numbers and Roots 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 β bi. If a quadratic equation with real coefficients has nonreal roots, those roots are complex conjugates. Helpful Hint When given one complex root, you can always find the other by finding its conjugate. Holt McDougal Algebra 2 2-5 Complex Numbers and Roots Example 5: Finding Complex Zeros of Quadratic Functions Find each complex conjugate. B. 6i A. 8 + 5i 8 + 5i 8 β 5i Write as a + bi. Find a β bi. Holt McDougal Algebra 2 0 + 6i Write as a + bi. 0 β 6i Find a β bi. β6i Simplify. 2-5 Complex Numbers and Roots Check It Out! Example 5 Find each complex conjugate. B. A. 9 β i 9 + (βi) 9 β (βi) 9+i C. β8i 0 + (β8)i 0 β (β8)i 8i Holt McDougal Algebra 2 Write as a + bi. Write as a + bi. Find a β bi. Find a β bi. Simplify. Write as a + bi. Find a β bi. Simplify. 2-5 Complex Numbers and Roots Lesson Quiz 1. Express in terms of i. Solve each equation. 2. 3x2 + 96 = 0 3. x2 + 8x +20 = 0 4. Find the values of x and y that make the equation 3x +8i = 12 β (12y)i true. 5. Find the complex conjugate of Holt McDougal Algebra 2 2-5 Complex Numbers and Roots Essential Question β’ How do you find complex zeros of quadratic functions? β’ Find the zeros of the related equation by completing the square and taking the square root of each side. Simplify the radical of the negative number by replacing with the imaginary unit i. Holt McDougal Algebra 2 2-5 Complex Numbers and Roots β’ Whatβs a mathematicianβs favorite dessert? β’ β1ce cream Holt McDougal Algebra 2