Warm-Up 1 (9.29.2014) 1) x2 – 7x + 12 = 0 (by factoring) Find the zeros of the following: 1) x2 – 7x + 12 = 0 (by factoring) 2) x2 + 4x – 16 = 0 (by quadratic Formula) 3) 2x2 + 8 = 0 (by sqrt method)

Essential Question (9.29.2014) How does the discriminant describe a quadratic function based on its solutions?

Ways to Solve Quadratic Equations and When to use each: Factor – Use if you can easily determine the factors Graphing – If you have access to a graphing calculator (Never preferred method anymore!) Solving for x – Use if there is no “b” value Quadratic Formula – ALWAYS works!

Types of Graphs: Graph the following Quadratics, complete a very rough sketch of the graph. (Just show the approximate x-intercepts from your calculator) 1. 2. 3.

How many times does each quadratic touch the x-axis? 1. 2. 3.

How many solutions does each quadratic have? The number of times a quadratic touches the x-axis is the number of solutions for the quadratic!!! 1. 2. 3.

Recall from the previous lesson, the quadratic formula: The Discriminant of the quadratic formula helps us to determine how many solutions a quadratic will have: The Discriminant is

What is the discriminate for each quadratic? 1. 2. 3.

Discriminant: If is a positive number there will be 2 real solutions If is zero there will be 1 solution. If is a negative number there will be 0 real solutions (2 nonreal #s)

EVERYTHING I NEED TO KNOW ABOUT the Discriminant! Value of the discriminant (b2 – 4ac) Number and type of roots What does the graph look like? b2 – 4ac is positive b2 – 4ac > 0 b2 – 4ac = 0 b2 – 4ac is negative b2 – 4ac < 0 Crosses x-axis twice 2 real roots Crosses x-axis once (x-intercept is the same as the vertex!) 1 real root No real roots (2 Imaginary) Never crosses the x-axis!

Given the following quadratics, use the discriminant to determine how many solutions it will have. 1. x2 – 6x + 11 = 2 2. 3x2 + 5x = 12   3. 3x2 + 48 = 0 4. x2 – 27 = 0 5. x2 + x + 1 = 0 6. x2 + 4x -1 = 0

Given the following graphs of a quadratic function: Determine the sign of the discriminant. Determine whether the solutions are real or imaginary.

Summary: What two methods of solving quadratics will ALWAYS give you an answer? How do you know when to use square roots to solve? What are 2 other terms for the x – intercepts of a quadratic function?