Quadratic Equations and Problem Solving Lesson 3.2

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

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?

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

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

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

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

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.

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 - ...

The Quadratic Formula Try it for the quadratic functions 4x2 - 7x + 3 = 0                           6x2 - 2x + 5 = 0

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

The Quadratic Formula Why does the second function give "non-real result?“ 6x2 - 2x + 5 = 0

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?

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

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

Assignment Lesson 3.2 Page 185 Exercises 1 – 49 odd