College Algebra Lesson A.6b Notes Name: ____________________________________ Solving Quadratic Equations with a Negative Discriminant We have discussed how a quadratic equation with a negative discriminant has no real number solution. For example, Cannot square root a negative in Ex 1: x 2 4 0 x 2 4 x 4 the real number system However, if we extend our number system to allow complex numbers, quadratic equations will ______ have a solution. Thus, if N is a positive real number, we define the __________________, denoted N , as N ______ Ex 2. Ex 3. Ex 4. 1 1i i 4 4i 2i 8 8i 2 2i Practice 1: Practice 2: 3 18 Let’s apply this to solving quadratic equations in the complex number system. Ex 5. Solve x 2 4 0 x2 4 0 x 2 4 x 4 x 2i x 4i Practice 3: Solve x 2 9 0 Now let’s look at quadratic equations where we must complete the square to solve. Ex 6. Solve x 2 4 x 8 0 in the complex number system x 2 4 x 4 8 4 (x 2) 2 4 x 2 4 x 2 2i x 2 2i Practice 4: Solve x 2 2x 4 0 in the complex number system College Algebra Lesson A.6b Notes Character of the Solutions of a Quadratic Equation In the complex number system, consider a quadratic equation ax 2 bx c 0 with real coefficients. 1. If b 2 4ac 0 , the equation ___________________________________. 2. If b 2 4ac 0 , the equation ___________________________________ . 3. If b 2 4ac 0 , the equation ___________________________________. Ex 7. Given x 2 4 x 8 from example 6, we can determine the character of its solution(s) b 2 4ac (4)2 4(1)(8) 16 0 Thus, two complex solutions This matches our solution of x 2 2i from example 6. Ex 8. Determine the character of the solutions for 9x 2 6x 1 0 , then solve for x. Thus, repeated real root b 2 4ac (6)2 4(9)(1) 0 We can factor to solve the equation 9x 2 6x 1 0 (3x 1)(3x 1) 0 x 13 , x 13 Repeated real root! Ex 9. Determine the character of the solutions for x 2 4 x 1 0 , then solve for x. Thus, two unequal real solutions b2 4ac (4)2 4(1)(1) 12 0 We can complete the square to solve the equation x 2 4x 4 1 4 (x 2)2 3 x 2 3 x 2 3 Two unequal real solutions! Practice 5: Determine the character of the solutions for x 2 6x 13 0, then solve for x.