Quadratic Equations and Problem Solving Lesson 3.2

2 Finding Zeros Often with quadratic functions f(x) = a*x 2 + bx + c we speak of “finding the zeros” This means we wish to find all possible values of x for which a*x 2 + bx + c = 0

3 Finding Zeros Another way to say this is that we are seeking the x- axis intercepts This is shown on the graph below Here we see two zeros – what other possibilities exist?

4 Zeros of the Quadratic Zeros are where the function crosses the x-axis Where y = 0 Consider possible numbers of zeros None (or two complex) One Two 

5 Factoring Given the function x 2 - 2x - 8 = 0 Factor the left side of the equation (x - 4)(x + 2) = 0 We know that if the product of two numbers a * b = 0 then either... a = 0 or b = 0 Thus either x - 4 = 0 ==> x = 4 or x + 2 = 0 ==> x = -2

6 Warning!! Problem... many (most) quadratic functions are NOT easily factored!! Example:

7 Completing the Square We work with a quadratic equation to make one side a perfect square Then we take the square root of both sides Not forgetting to use both the + and - values of the right side of the equation

8 The Quadratic Formula We can use completing the square with the general equation ax 2 + bx + c = 0. Once this is done, we can use the formula for any quadratic function.

9 The Quadratic Formula It is possible to create two functions on your calculator to use the quadratic formula. quad1 (a,b,c) which uses the -b +... quad2 (a,b,c) which uses the -b -...

10 The Quadratic Formula Try it for the quadratic functions 4x 2 - 7x + 3 = 0 6x 2 - 2x + 5 = 0

11 The Quadratic Formula 4x 2 - 7x + 3 = 0

12 The Quadratic Formula Why does the second function give "non-real result?“ 6x 2 - 2x + 5 = 0

13 The Discriminant Consider the expression under the radical in the quadratic formula This is known as the discriminant What happens when it is Positive and a perfect square? Positive and not a perfect square? Zero Negative?

14 Graphical Solution Given Manipulate the equation to be equal to zero Specify this as a function of x on Y= screen Graph and note zeros Use F5 menu

15 Numeric Solution Given As before … Manipulate the equation to be equal to zero Specify this as a function of x on Y= screen Now go to the Table, use ♦Y Look for x-value where y-values go from negative to positive Use setup, F2 to change start and increment to "zoom in" on the numeric answer

16 Assignment Lesson 3.2 Page 200 Exercises 1 – 77 EOO