Section 11.3 Studying Solutions of Quadratic Equations Idea of 11.3 ● ● ● ● Section 11.2 showed us how to find the answers. Section 11.3 we show us what type of answers we will have. The types of answers are – Repeated rational number – Two distinct rational numbers. – Two irrational conjugates – Two complex conjugates We will determine the type of answer by looking at the determinant. The determinant is b² – 4ac. Repeated Rational Number ● The answers will be a repeated rational number if b² - 4ac = 0 Example ● Determine the types of answers given 4x² – 12x + 9 = 0 ● Therefore a = 4 b = -12 c = 9 ● b² – 4ac ● (-12)² – 4(4)(9) = 144 – 144 = 0 ● Therefore this equation will have one rational number that will be repeated. Two distinct rational numbers. ● The answers will be two distinct rational numbers if b² - 4ac is positive and a perfect square. Example ● Determine the types of answers given 4m² + 7m = 0 ● Therefore a = 4 b = 7 c = 0 ● b² – 4ac ● (7)² – 4(4)(0) = 49 – 0 = 49 ● 49 is positive and a perfect square, therefore this equation will have two distinct rational numbers. Two Irrational Conjugates ● The answers will be two irrational conjugate numbers if b² - 4ac is positive but not a perfect square Example ● Determine the types of answers given 10t² - t - 2 = 0 ● Therefore a = 10 b = -1 c = -2 ● b² – 4ac ● (-1)² – 4(10)(-1) = 1 + 40 = 41 ● 41 is positive, but is not a perfect square, therefore this equation will have two distinct irrational conjugates. Two Complex Conjugates ● The answers will be two complex conjugate numbers if b² - 4ac is negative Example ● Determine the types of answers given 3x² + 5 = -7x ● Therefore a = 3 b = 7 c = 5 ● b² – 4ac ● (7)² – 4(3)(5) = 49 – 60 = -11 ● -11 is a negative number, therefore this equation will have two complex conjugates. How to find the equation ● How to find the equation given the solutions. – Start with the answers equal to the variable. – Write them as factors. – Combine the factors through multiplication. – FOIL – Simplify – Equation Graphs and Discriminants ● The discriminant can help you create the graph. ● How? Example ● Find the equation given the answers -6 and 3 ● x = -6 or x = 3 ● x + 6 = 0 or x – 3 = 0 ● (x + 6)(x – 3) = 0*0 ● X² – 3x + 6x - 18 = 0 ● X² + 3x - 18 = 0 Example ● Find the equation given the answers 3, -1, 0 Homework ● Section 11.3 ● # 7, 9, 13, 17, 29, 35, 53, 45, 56