5-5 Complex Numbers and Roots TEKS 2A.2.B Foundations for functions: use complex numbers to describe the solutions of quadraic equations. Objectives Why learn this? Define and use imaginary Complex numbers can be used to and complex numbers. describe the zeros of quadratic Solve quadratic equations functions that have no real with complex roots. zeros. (See Example 4.) Vocabulary You can see in the graph of imaginary unit f (x) = x 2 + 1 below that f has no real imaginary number zeros. If you solve the corresponding complex number = 2 + real part equation 0 x 1, you find that = ±√ - imaginary part x 1 , which has no real solutions. Andrew Toos/CartoonResource.com complex conjugate However, you can find solutions if you define Þ the square root of negative numbers, which is why imaginary numbers were invented. The v­Ý®ÊÊÝÓÊ Ê£ √- Also 2A.2.A, 2A.5.E, imaginary unit i is defined as 1 . You can use Ý the imaginary unit to write the square root of any 2A.6.A, 2A.6.B, 2A.8.A, ÊÝÌiÀVi«Ìà 2A.8.D negative number. Imaginary Numbers WORDS NUMBERS ALGEBRA An imaginary number is the square √- 1 = i If b is a positive root of a negative number. real number, Imaginary numbers can be written √- 2 = √- 1 √2 = i √2 then √- b = i √b in the form bi, where b is a real √- = √- √ = √- 2 = number and i is the imaginary unit. 4 1 4 2i and b bi. 2 2 The square of an imaginary number ( √- 1 ) = i 2 = -1 ( √- b ) = -b is the original negative number. EXAMPLE 1 Simplifying Square Roots of Negative Numbers Express each number in terms of i. A 3 √- 16 B - √- 75 3 √ (16) (-1) Factor out -1. - √ (75) (-1) Factor out -1. 3 √16 √- 1 Product Property - √75 √- 1 Product Property 3 · 4 √- 1 Simplify. - √25 √3 √- 1 Product Property 12 √- 1 Multiply. -5 √3 √- 1 Simplify. 12i Express in terms -5 √3 i = -5i √3 Express in terms of i. of i. Express each number in terms of i. 1a. √- 12 1b. 2 √- 36 1c. - _1 √- 63 3 350 Chapter 5 Quadratic Functions aa207se_c05l05_0350_0355.indd207se_c05l05_0350_0355.indd 335050 112/14/052/14/05 11:52:32:52:32 PPMM EXAMPLE 2 Solving a Quadratic Equation with Imaginary Solutions Solve each equation. A x 2 = -81 B 3 x 2 + 75 = 0 x = ± √- 81 Take square 3 x 2 = -75 Add -75 to both sides. roots. x 2 = -25 Divide both sides by 3. x = ±9i Express in terms of i. = ±√- x 25 Take square roots. Check x = ±5i Express in terms of i. −−−−−−−−− x 2 = -81 −−−−−−−−− x 2 = -81 2 + = (9i) 2 -81 (-9i) 2 -81 Check −−−−−−−−−−−− 3 x 75 0 3 (±5i) 2 + 75 0 81i 2 -81 81i 2 -81 ( ) 2 + 81 (-1) -81 ✔ 81 (-1) -81 ✔ 3 25 i 75 0 (- ) + ✔ 75 1 75 0 Solve each equation. 2a. x 2 = -36 2b. x 2 + 48 = 0 2c. 9 x 2 + 25 = 0 A complex number is a number «iÝÊ ÕLiÀÃÊ ο that can be written in the form a + bi, where a and b are real ÎÊ ÊÇ ÊÊÊÊÎÊ ÊÊÚÚÊÓÊÊÊÊ ÊÊÊÊ{ÊÊ Î numbers and i = √- 1 . The set of real numbers is a subset of the set ,i> >}>ÀÞ of complex numbers ¼. ÕLiÀÃÊ ώ ÕLiÀà ÚÚ£ÊÊÊÊÊÊÊÊ£°ÇÎÊÊÊÊäÊÊÊÊû ÊÊÊÊÎÊÊÊÊx Every complex number has a real Ó Ü part a and an imaginary part b. °ÊÈÊÊÊÊÊÊÊȖÓÊÊе ÊȖее ÇÊÊ Real part Imaginary part Real numbers are complex numbers where b = 0. Imaginary numbers are complex numbers where a = 0 and b ≠ 0. These are sometimes called pure imaginary 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 3x - 5i = 6 - ( 10y) i true. Real parts 3x - 5i = 6 - (10y) i Imaginary parts 3x = 6 Equate the real parts. -5 = -( 10y) Equate the imaginary parts. x = 2 Solve for x. _1 = y Solve for y. 2 Find the values of x and y that make each equation true. 3a. 2x - 6i = -8 + (20y) i 3b. -8 + (6y) i = 5x - i √ 6 5- 5 Complex Numbers and Roots 351 aa207se_c05l05_0350_0355.indd207se_c05l05_0350_0355.indd 335151 112/14/052/14/05 11:52:41:52:41 PPMM EXAMPLE 4 Finding Complex Zeros of Quadratic Functions Find the zeros of each function. 2 2 A f (x) = x - 2x + 5 B g (x) = x + 10x + 35 2 2 x - 2x + 5 = 0 Set equal to 0. x + 10x + 35 = 0 2 2 x - 2x + = -5 + Rewrite. x + 10x + = -35 + 2 2 - + = - + Add (__b ) . 2 + + = - + x 2x 1 5 1 2 x 10x 25 35 25 (x - 1) 2 = -4 Factor. (x + 5) 2 = -10 x - 1 = ± √- 4 Take square roots. x + 5 = ± √- 10 x = 1 ± 2i Simplify. x = -5 ± i √10 Find the zeros of each function. 4a. f (x) = x 2 + 4x + 13 4b. g (x) = x 2 - 8x + 18 The solutions -5 + i √ 10 and -5 - i √ 10 in Example 4B are related. These When given one solutions are a complex conjugate pair. Their real parts are equal and their complex root, you imaginary parts are opposites. The complex conjugate of any complex number can always find the a + bi is the complex number a - bi. other by finding its conjugate. If a quadratic equation with real coefficients has nonreal roots, those roots are complex conjugates. EXAMPLE 5 Finding Complex Conjugates Find each complex conjugate. A 2i - 15 B -4i -15 + 2i Write as a + bi. 0 + (-4) i Write as a + bi. -15 - 2i Find a - bi. 0 - (-4) i Find a - bi. 4i Simplify. Find each complex conjugate. 5a. 9 - i 5b. i + √3 5c. -8i THINK AND DISCUSS 1. Given that one solution of a quadratic equation is 3 + i, explain how to determine the other solution. 2. Describe a number of the form a + bi in which a ≠ 0 and b = 0. Then describe a number in which a = 0 and b ≠ 0. Are both numbers complex? Explain. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box or «iÝÊ ÕLiÀà oval, give a definition and examples of ,i>Ê >}>ÀÞÊ each type of number. ÕLiÀà ÕLiÀà 352 Chapter 5 Quadratic Functions aa207se_c05l05_0350_0355.indd207se_c05l05_0350_0355.indd 335252 112/14/052/14/05 11:52:44:52:44 PPMM.