Solving Quadratic Functions Lesson 5.5b

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

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?

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

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 ax 2 + 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  4x 2 - 7x + 3 = 0  6x 2 - 2x + 5 = 0

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

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

Assignment Lesson 5.5b Page 220 Exercises 27 – 35 odd