1. Solving Quadratic Equations by the Quadratic Formula

2. What does it mean to “solve”?
When you are asked to “solve” a quadratic, it is asking you to find the roots.

3. THE QUADRATIC FORMULA

4. What do we do with it?
Step 1: plug 0 in for y, and get the 0 by itself.
Step 2: Just identify a, b, and c then substitute into the formula.

5. Example 𝑦= 𝑥 2 +4𝑥+3 a=1. b=4. c=3 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎
𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎
𝑥= −4± −4∙1∙3 2∙1

6. Example 1 continued
𝑥= −4± 4 2
𝑥= − 𝑎𝑛𝑑 𝑥= −4−2 2
x=-1 and x=-3

7. WHY USE THE QUADRATIC FORMULA?
The quadratic formula allows you to solve ANY quadratic equation, even if you cannot factor it.

8. Definition
Definition:Discriminant is the piece under the radical: b2 – 4ac

9. WHY IS THE DISCRIMINANT IMPORTANT?
The discriminant tells you the number and types of answers
(roots) you will get. The discriminant can be +, –, or 0
Since the discriminant is under a radical, think about what it means if you have a positive or negative number or 0 under the radical.

10. WHAT THE DISCRIMINANT TELLS YOU!
Value of the Discriminant
Nature of the Solutions
Negative
2 imaginary solutions
Zero
1 Real Solution
Positive
2 Real Solutions

11. Refresh What is an imaginary number?
When you have a square root of a negative number.
Called i.
−1 =𝑖

12. Example with imaginary answer:
𝑥 2 −2𝑥+5=0 Try solving on your own…

13. 𝑥 2 −2𝑥+5=0 𝑥= −(−2)± (−2) 2 −4∙1∙5 2∙1 Plug in a,b and c
𝑥= 2± −16 2∙1 Simplify under the radical
𝑥= 2±𝑖 Factor out the -1, it becomes i.
𝑥= 2±𝑖∙ Take square root
𝑥=1+2𝑖 𝑎𝑛𝑑 𝑥=1−2𝑖 Divide by 2.

14. You try! Start with 1 and 4, then go on to the others.

15. Write a quadratic in general from:
and the x-intercepts are 5 and 2.
What can we do?

16. Solution
Write it factored form (because that is what information we have.
𝑦=(𝑥−5)(𝑥−2)
Then foil to get it in general
𝑦= 𝑥 2 −7𝑥−10

17. Homework: This is a massive assignment!
7.4 Worksheet
1 a-c
2 all
3 a-f
4 a-d
5 a-c