Algebra Unit: 05 Lesson: 0 Complex Numbers All the quadratic equations solved to this point have had two real solutions or roots. In some cases, solutions involved a double root, but there were always two solutions. Study the quadratic equation 0 = x + x +. Can this equation be factored in order to find the solutions or roots? Can the zeros be found from the graph of the function y = x + x +? Explain. There are still two solutions to this quadratic equation. However, they are not real numbers. Instead, the solutions involve numbers that are imaginary. The imaginary number unit is designated as i. There are two ways that we define i. Words Symbols i is the number you can square to get an answer of -1. Or, i is the square root of -1 Square roots of negative numbers can be simplified using i : If x is a non-negative real number, then x = ( 1) x = 1 x = i x When you have a negative number under a radical, you can take the negative out as an i. Example Problems: Simplify the following complex numbers by writing them in terms of i. A) - 49 = Check: ( ) = = = -49 B) - 11= Check: ( ) = = = -11 C) 17-4 = D) - 3 = E) 1 + - 36 01, TESCCC 09/06/1 page 1 of 6
Algebra Unit: 05 Lesson: 0 A complex number is the sum of a real number and an imaginary number, usually written in the form a + bi. Real part Imaginary part Write the following numbers in a + bi form, and identify the real and imaginary parts. Simplify first if necessary. F) 5 G) 5i H) 3 5i I) 9 + - 144 6 J) 3 + 5i 7 + 8i K) (6 + i)(6 i) The conjugate of a complex number has the same real Complex Conjugates part but an opposite imaginary part. (For example, + 3i is the complex conjugate of - 3i.) a + bi and a - bi * When complex conjugates are added or multiplied the result will have an imaginary part of 0 i. Tell the conjugate of each complex number. Number Conjugate Number Conjugate Number Conjugate L) 5 N) -3 + 6i P) 1 - i M) 5i O) -1 5i Q) 5 + i 3 3 5 Solving Equations with Complex Numbers Some quadratic equations can be solved by isolating the variable and taking the square root of both sides. Sample: x = - 100 x = ± - 100 x = ± 10i R) Solve 6x + 96 = 0 Why are the solutions complex conjugates? 01, TESCCC 09/06/1 page of 6
Algebra Unit: 05 Lesson: 0 Describe the complex numbers below by filling in the table. Chart A Simplified Real Part Imaginary Part Complex Conjugate 1) - 11 ) -56 3) 85 + - 169 16 4) - - 49 5) - 5 6) 3 14i 9 + 5i 7) (5 + i) Solve the following quadratic equations and describe the quadratic solutions. 8) x + 50 = 0 9) - 3x - 8 = 100 10) What is the quadratic equation with solutions of ± 7i? 01, TESCCC 09/06/1 page 3 of 6
Algebra Unit: 05 Lesson: 0 Part III: Quadratic Formula Any quadratic equation including those with imaginary or complex solutions can be solved with the quadratic formula. Quadratic Formula If ax + bx + c = 0, and a 0, then - x = b ± b a - 4ac Steps: Get one side of the equation equal to zero. Identify the values of a, b, and c. Substitute these values into the formula. Simplify to obtain the two solutions. From the quadratic formula, b - 4ac is called the discriminant. Because this part of the formula lies under the radical, it determines the type of solutions the quadratic equation will have. If Then the equation has Sample Previous Methods x + 6x + 9 = 0 x + 6x + 9 = 0 = 0 One rational double root a = 1, b = 6, c = 9 ( x + 3)( x + 3) = 0 (6) - 4(1)(9) = 0 x = -3 (double root) > 0, x + x 3 = 0 x + x - 3 = 0 and b - 4ac is a (x+ 3)( x 1) = 0 Two rational roots a =, b = 1, c = - 3 perfect square 3 (1, 4, 9, 16, 5, etc.) (1) - 4()( - 3) = 5 x =, x = 1 > 0, x - 10x + 3 = 0 Will not factor and b - 4ac is a Graphs can provide Two irrational roots a = 1, b = - 10, c = 3 non-square approximations only (, 5, 8, 1, etc.) (- 10) - 4(1)(3) = 8 (Here, x = 5 ± ) Two complex roots x - 4x + 40 = 0 Will not factor < 0 (Conjugates involving a = 1, b = - 4, c = 40 Graphs cannot imaginary numbers) (- 4) - 4(1)(40) = - 144 provide solutions (Here, x = ± 6i ) Example Problems For each problem, a) Write the equation in the form ax + bx + c = 0. b) Find the discriminant, and use it to describe the nature of the roots of the equation. c) Complete the quadratic formula to find the solutions. 1) 3x + 7x = - 01, TESCCC 09/06/1 page 4 of 6
Algebra Unit: 05 Lesson: 0 ) x - x = x - 3) x = 14x - 58 4) Given a quadratic equation with rational coefficients and a root of 3 + 5i d) What is the representative function? 5) Given a quadratic equation with rational coefficients and a root of - + 8i d) What is the representative function? 01, TESCCC 09/06/1 page 5 of 6
Algebra Unit: 05 Lesson: 0 Practice Problems For each problem, Use the discriminant to describe the nature of the roots of the equation. Complete the quadratic formula to find the solutions. 1) x 8x+ 1 = 0 ) 3x + x = 10 3) x + 5 = 10x 4) 10x + x+ 5 = 0 5) x + 6 = 8x 6) x + x 14 = 11 7) Given a quadratic equation with rational coefficients and a root of 3 i 8) Given a quadratic equation with rational coefficients and a root of 5 i 01, TESCCC 09/06/1 page 6 of 6