6 THE QUADRATIC FORMULAWhen you solve using completing the square on the general formula you get:This is the quadratic formula!Just identify a, b, and c then substitute into the formula.
7 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
8 What is the discriminant? The discriminant is the expressionb2 – 4ac.We represent the discriminant with DThe value of the discriminant can be usedto determine the number and type of rootsof a quadratic equation.
9 How have we previously used the discriminant? We used the discriminant to determinewhether a quadratic polynomial couldbe factored.If the value of the discriminant for aquadratic polynomial is a perfect square,the polynomial can be factored.
10 Let’s put all of that information in a chart. Value of DiscriminantType andNumber of RootsSample Graphof Related FunctionD > 0,D is a perfect squareD NOT a perfect squareD = 0D < 0
11 During this presentation, we will complete a chart that shows how the value of the discriminantrelates to the number and type of roots of aquadratic equation.Rather than simply memorizing the chart, thinkAbout the value of b2 – 4ac under a square rootand what that means in relation to the roots ofthe equation.
12 Solve These…Use the quadratic formula to solve eachof the following equations?x2 – 5x – 14 = 02x2 + x – 5 = 0x2 – 10x + 25 = 04x2 – 9x + 7 = 0
13 Let’s evaluate the first equation. x2 – 5x – 14 = 0What number is under the radical whensimplified?81The discriminant is 81; which is a perfect squareWhat are the solutions of the equation?–2 and 7 which are both rational numbers.
14 If the value of the discriminant is positive, the equation will have 2 real roots.If the value of the discriminant is aperfect square, the roots will be rational.
15 Let’s look at the second equation. 2x2 + x – 5 = 0What number is under the radical whensimplified?41 (which is not a perfect square.)What are the solutions of the equation?Both solutions are irrational
16 If the value of the discriminant is positive, the equation will have 2 real roots.If the value of the discriminant is a NOTperfect square, the roots will be irrational.
17 Now for the third equation. x2 – 10x + 25 = 0What number is under the radical whensimplified?What are the solutions of the equation?5 (double root)
18 If the value of the discriminant is zero, the equation will have 1 real, root; it willbe a double root.If the value of the discriminant is 0, theroots will be rational.
19 Last but not least, the fourth equation. 4x2 – 9x + 7 = 0What number is under the radical whensimplified?–31What are the solutions of the equation?There are no real solutions. The solution is imaginary.
20 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.
21 Let’s put all of that information in a chart. Value of DiscriminantType andNumber of RootsSample Graphof Related FunctionD > 0,D is a perfect square2 real,rational rootsD NOT a perfect squareIrrational rootsD = 01 real, rational root(double root)D < 02 complex roots(complex conjugates)
22 Try These. For each of the following quadratic equations, Find the value of the discriminant, andDescribe the number and type of roots.x2 + 14x + 49 = x2 + 8x + 11 = 02. x2 + 5x – 2 = x2 + 5x – 24 = 0
24 WHY IS THE DISCRIMINANT IMPORTANT? The discriminant tells you the number and types of answers(roots) you will get. The discriminant can be +, –, or 0which actually tells you a lot! Since the discriminant isunder a radical, think about what it means if you have apositive or negative number or 0 under the radical.
25 WHAT THE DISCRIMINANT TELLS YOU! Value of the DiscriminantNature of the SolutionsNegative2 imaginary solutionsZero1 Real SolutionPositive – perfect square2 Reals- RationalPositive – non-perfect square2 Reals- Irrational
26 Example #1 a=2, b=7, c=-11 Discriminant = Discriminant = 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)1.a=2, b=7, c=-11Discriminant =Value of discriminant=137Positive-NON perfect squareNature of the Roots – 2 Reals - IrrationalDiscriminant =
27 Example #1- continuedSolve using the Quadratic Formula
28 Solving Quadratic Equations by the Quadratic Formula Try the following examples. Do your work on your paper and then check your answers.