1. Let’s Review the Quadratic Formula & Completing the Square!!

2. You must have a QUADRATIC before solving!
Quadratic Formula
You must have a QUADRATIC before solving!

3. Quadratic Formula: Make sure each variable is in parenthesis.
Solve for the discriminant. JUST the part UNDER the radical.
Split up the fractions.
Simplify each fraction.

4. Example A:

5. Completing the Square:
Put in standard form.
Move “c” to the other side.
Add to both sides of the equation .
Write as a
Take the square root of both sides
Simplify

6. Example B:

7. You Try!
1) )

8. What about…

9. IRRATIONAL
How do I find the x-intercepts of a polynomial equation that will not factor?

10. IRRATIONAL ROOTS
This will occur when you get a polynomial after synthetic division that CANNOT be factored!
But you MUST get it down to a quadratic!
Might need to use the Quadratic Formula:

11. f(x) = x4 + 3x3 – 5x2 – 15x 1 3 -5 -15 -3 x = 0 , -3 1 3 -5 -15 -3 15
What if it doesn’t factor to begin with or you aren’t sure?
Find a root to begin synthetic division using your calculator, and get it down to a quadratic!
f(x) = x4 + 3x3 – 5x2 – 15x
x = 0
, -3 1 3 -5
-15 -3 1 3 -5
-15 -3 15 1 -5

12. Find all of the roots of: g(x) = x4 + 2x3 – 5x2 – 4x + 6
(-3 and 1 using the calculator)
Find all of the roots of:
g(x) = x4 + 2x3 – 5x2 – 4x + 6
ANSWER: -3, 1, ,

13. Find all of the roots of: g(x) = 4x3 – 16x2 + 11x + 3
(x=3 using the calculator)
Find all of the roots of:
g(x) = 4x3 – 16x2 + 11x + 3
ANSWER: 3, ,

14. Find all of the zeros of: g(x) = 3x3 – 22x2 + 32x + 32
(x=4 using the calculator)
Find all of the zeros of:
g(x) = 3x3 – 22x2 + 32x + 32
ANSWER: 4, 4 ,

15. Find all of the roots of: g(x) = x3 – x2 – 11x + 3
(x=-3 using the calculator)
Find all of the roots of:
g(x) = x3 – x2 – 11x + 3
ANSWER: -3, ,

16. Work and sketch on a separate piece of paper.
Homework!!
Work and sketch on a separate piece of paper.