1. Quadratic Equations and Problem Solving
Lesson 3.2

2. Finding Zeros
Often with quadratic functions     f(x) = a*x2 + bx + c   we speak of “finding the zeros”
This means we wish to find all possible values of x for which    a*x2 + 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 x2 - 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
ax2 + 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
4x2 - 7x + 3 = 0                          
6x2 - 2x + 5 = 0

11. The Quadratic Formula
4x2 - 7x + 3 = 0  

12. The Quadratic Formula
Why does the second function give "non-real result?“
6x2 - 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 … Now go to the Table, use ♦Y
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 185
Exercises 1 – 49 odd