Ch3/4 Lesson 9 The Discriminant Nature of the ROOTS

I) Nature of the ROOTS The “Nature of the Roots” refers to the “Number of X-intercepts” that a Quadratic Function will have A Quadratic function can have either 2 distinct roots (2 different x intercepts) 2 equal roots (1 distinct root, double root) No real roots (no x intercepts) Use the “Discriminant Formula” to find the Nature of the roots:

II) Discrminant There will be 2 distinct roots x y -2 -1 1 2 x - There will be 2 distinct roots x y -1 1 2 3 4 x 4 There will be a double root/ 2 equal roots x y -2 -1 1 2 4 x There will be no x intercepts/ No real roots

Ex: Determine the nature of the roots for each equation: (Do not Solve) Discriminant is negative  There are no real roots Discriminant is positive  There are 2 real roots

II) Review Solving Inequalities: Whenever you divide or multiply by a negative number, the inequality sign will switch sides

Solving Inequalities with Squares: When you have inequalities with squares, isolate the square and then square root both sides Draw a number line, draw the roots on the number line You will get three different domains on the number line Pick a value from each domain and check if it satisfies the inequality Domains that satisfy the equality will be your solution

Example: Solve the Following Inequalities:

Practice: Solve the Following Inequalities:

III) Nature of Roots (Solving For “K”) Ex: For what values of “k” does the equation have two different real roots? Since the equation has 2 different real roots, The discriminant must be greater than zero IF “K” is less than -8 or greater than 8, the quadratic equation will have two X-intercepts

IF K equals 6 or -6, the equation will have a double root! Practice: For what values of “k” does the equation have a) Two equal roots b)No real roots a) Since the equation has 2 equal roots, The discriminant must be equal to zero b) No real roots discriminant must be less than zero IF K equals 6 or -6, the equation will have a double root! IF K is between 6 or -6, the equation will not have any roots

If “k” is greater than 2.5, the function will have no roots Practice: For what values of “k” does the equation have a) Two different roots b)No real roots b) the discriminant is less than zero a) the discriminant is greater than zero If “k” is greater than 2.5, the function will have no roots If “k” is less than 2.5, the function will have two roots

Homework: Assignment 4.5