1. The Roots of a Quadratic Equation

2. What are the zeros/roots? They are where the arms of a parabola cross the x-axis, and y is zero. There can be two zeros, one zero, or no zeros, depending on where the parabola is on the graph.

3. Methods to finding Zeros: 1.) Quadratic formula (need standard form) 2.) Factor into zeros form 3.) Graphing calculator (or graphing by hand, but let’s face it, that’s a waste of time)

4. Quadratic Formula:

5. Ex.1) 0 = -2x^2 +12x – 18 a = -2, b = 12, c = -18 X= -12 + 12^2 – 4(-2)(-18) 2(-2) X= -12 + 144 – 144 X= -12 + 0 -4 -4 X= -12 -4 X= 3

6. Ex.2) 0 = 2x^2 + 6x – 8 a = 2, b = 6, c= -8 X= -6 + 6^2 – 4(2)(-8) 2(2) X= -6 + 36 + 64 X= -6 + 100 4 4 X= -6 -10 OR X= -6 + 10 4 4 X= -4 OR X= 1

7. Ex.3) 0= x^2 – 4x + 7 a = 1, b = -4, c = 7 X= -(-4) + (-4)^2 – 4(1)(7) 2(1) X= 4 + 16 – 28 2 X= 4 + -12 2 You can’t square root a negative, therefore there are no zeros! 

8. Factoring Method: -To factor into zeros form, you must put into standard form first -Zeros form looks like this: f(x)= a (x-t) (x-s) The x values are the zeros.

9. Ex.1) F(x) = -2x^2 +12x – 18 F(x)= -2 (x^2 – 6x + 9) = -2 (x^2 – 3x + 3x + 9) = -2[ x ( x – 3) - 3 ( x - 3 ) ] = -2 (x-3)^2 Because the x value is the same thing, there is only ONE zero.

10. Ex.2) F(x) = 2x^2 + 6x – 8 = 2 (x^2 + 3x - 4) = 2 (x^2 - x + 4x – 4) = 2[ x (x – 1) + 4 (x – 1) ] = 2 (x – 1) (x + 4) Because there are two x values, there are also two zeros.

11. Ex.3) F(x) = x^2 – 4x + 7 Noooooot factorable, therefore there are no zeros! 

12. Graphing Calculator

13. khanacademy.org wolframalpha.com