The Discriminant Check for Understanding – Given a quadratic equation use the discriminant to determine the nature of the roots. Terms we need to know: 1) Quadratic Formula 2) Real number system 3) Rational numbers 4) Irrational numbers 5) Perfect squares 6) Complex numbers 7) Imaginary numbers Examples of Rational and Irrational numbers Rational Irrational a)2 a) 24 12 b) b) 휋 24 3 c) 25 c) 10 d) 3 8 Most common irrational numbers involve the √ symbol. Many square roots, cube roots, etc. are irrational. Let’s Practice • Record your answer for each Complex Number System Reals Imaginary Rationals i, 2i, -3-7i, etc. (fractions, decimals) Integers (…, -1, -2, 0, 1, 2, …) Whole Irrationals (no fractions) (0, 1, 2, …) pi, e Natural (1, 2, …) THE QUADRATIC FORMULA 1. When you solve using completing the square on the general formula2 you get: ax bx c 0 b b2 4 ac x 2a 2. This is the quadratic formula! 3. Just identify a, b, and c then substitute into the formula. WHY USE THE QUADRATIC FORMULA? The quadratic formula allows you to solve ANY quadratic equation, even if you cannot factor it. An important piece of the quadratic formula is what’s under the radical What is the discriminant? The discriminant is the expression b2 – 4ac. We represent the discriminant with D The value of the discriminant can be used to determine the number and type of roots of a quadratic equation. How have we previously used the discriminant? We used the discriminant to determine whether a quadratic polynomial could be factored. If the value of the discriminant for a quadratic polynomial is a perfect square, the polynomial can be factored. Let’s put all of that information in a chart. Type and Sample Graph Value of Discriminant Number of Roots of Related Function D > 0, D is a perfect square D > 0, D NOT a perfect square D = 0 D < 0 During this presentation, we will complete a chart that shows how the value of the discriminant relates to the number and type of roots of a quadratic equation. Rather than simply memorizing the chart, think About the value of b2 – 4ac under a square root and what that means in relation to the roots of the equation. Solve These… Use the quadratic formula to solve each of the following equations? 1. x2 – 5x – 14 = 0 2. 2x2 + x – 5 = 0 3. x2 – 10x + 25 = 0 4. 4x2 – 9x + 7 = 0 Let’s evaluate the first equation. x2 – 5x – 14 = 0 What number is under the radical when simplified? 81 The discriminant is 81; which is a perfect square What are the solutions of the equation? –2 and 7 which are both rational numbers. If the value of the discriminant is positive, the equation will have 2 real roots. If the value of the discriminant is a perfect square, the roots will be rational. Let’s look at the second equation. 2x2 + x – 5 = 0 What number is under the radical when simplified? 41 (which is not a perfect square.) What are the solutions of the equation? Both solutions are irrational 1 41 4 If the value of the discriminant is positive, the equation will have 2 real roots. If the value of the discriminant is a NOT perfect square, the roots will be irrational. Now for the third equation. x2 – 10x + 25 = 0 What number is under the radical when simplified? 0 What are the solutions of the equation? 5 (double root) If the value of the discriminant is zero, the equation will have 1 real, root; it will be a double root. If the value of the discriminant is 0, the roots will be rational. Last but not least, the fourth equation. 4x2 – 9x + 7 = 0 What number is under the radical when simplified? –31 What are the solutions of the equation? There are no real solutions. The solution is imaginary. 9 i 31 8 If the value of the discriminant is negative, the equation will have 2 complex roots; they will be complex conjugates. Not to panic if you don’t recognize these above terms… we will cover them in the next lesson. Let’s put all of that information in a chart. Type and Sample Graph Value of Discriminant Number of Roots of Related Function D > 0, 2 real, D is a perfect square rational roots D > 0, 2 real, D NOT a perfect square Irrational roots 1 real, rational root D = 0 (double root) 2 complex roots D < 0 (complex conjugates) Try These. For each of the following quadratic equations, a) Find the value of the discriminant, and b) Describe the number and type of roots. 1. x2 + 14x + 49 = 0 3. 3x2 + 8x + 11 = 0 2. x2 + 5x – 2 = 0 4. x2 + 5x – 24 = 0 The Answers 1. x2 + 14x + 49 = 0 3. 3x2 + 8x + 11 = 0 D = 0 D = –68 1 real, rational root 2 complex roots (double root) (complex conjugates) 2 2. x + 5x – 2 = 0 4. x2 + 5x – 24 = 0 D = 33 D = 121 2 real, irrational roots 2 real, rational roots WHY IS THE DISCRIMINANT IMPORTANT? The discriminant tells you the number and types of answers (roots) you will get. The discriminant can be +, –, or 0 which actually tells you a lot! Since the discriminant is under a radical, think about what it means if you have a positive or negative number or 0 under the radical. WHAT THE DISCRIMINANT TELLS YOU! Value of the Discriminant Nature of the Solutions Negative 2 imaginary solutions Zero 1 Real Solution Positive – perfect square 2 Reals- Rational Positive – non-perfect 2 Reals- Irrational square Example #1 Find the value of the discriminant and describe the nature of the roots (real,imaginary, rational, irrational) of each quadratic equation. Then solve the equation using the quadratic formula) 2xx2 7 11 0 a=2, b=7, c=-11 1. Discriminant = b2 4 ac Value of discriminant=137 2 (7) 4(2)( 11) Positive-NON perfect square 49 88 Nature of the Roots – Discriminant = 137 2 Reals - Irrational Example #1- continued Solve using the Quadratic Formula 2xx2 7 11 0 ab2, 7,c 11 b b2 4 ac 2a 7 72 4(2)( 11) 2(2) 7 137 2 Reals - Irrational 4 Solving Quadratic Equations by the Quadratic Formula Try the following examples. Do your work on your paper and then check your answers. 2 1. 9,7 1. xx2 63 0 2.(6, 14) 2. xx2 8 84 0 3. 3,8 3. xx2 5 24 0 2 73i 4. xx7 13 0 4. 2 5. 3xx2 5 6 0 5i 47 5. 6.