1. 9.5 Quadratic Formula

2. What We Will Learn Solve using Quadratic Formula
Interpret discriminant
Choose methods for solving quadratics

3. Essential Question
How can you use the appropriate method to solve quadratic equations?

4. Needed Vocab Quadratic Formula: 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎
Discriminant: 𝑏 2 −4𝑎𝑐

5. Ex. 1 Using Quadratic Formula
Solve 2𝑥 2 −5𝑥+3=0
𝑎=2, 𝑏=−5, 𝑐=3
𝑥= 5± (−5) 2 −4(2)(3) 2(2)
𝑥= 5± 25−24 4
𝑥= 5± 1 4
𝑥= , 𝑥= 5−1 4
𝑥= 3 2 , 𝑥=1
Steps
1. get = to 0
2. plug in a, b, c
3. do pemdas

6. Ex. 1 Cont. Solve −3𝑥 2 +2𝑥+7=0 𝑎=−3, 𝑏=2, 𝑐=7
𝑥= −2± −4(−3)(7) 2(−3)
𝑥= −2± −6
𝑥= −2± 88 −6
𝑥= − −6 , 𝑥= −2− 88 −6
𝑥=−1.23, 𝑥=1.90
Your Practice
Solve 2𝑥 2 +9𝑥+7=3
−3 −3
2𝑥 2 +9𝑥+4=0
𝑎=2, 𝑏=9, 𝑐=4
𝑥= −9± −4(2)(4) 2(2)
𝑥= −9± 81−32 4
𝑥= −9±
𝑥= −9+7 4 , 𝑥= −9−7 4
𝑥=− 1 2 , 𝑥=−4

7. Exs. 3/4 Finding Number of Real Solutions and Number of x-intercepts
Use discriminant
𝑏 2 −4𝑎𝑐
If 𝑏 2 −4𝑎𝑐>0, then If 𝑏 2 −4𝑎𝑐=0, then If 𝑏 2 −4𝑎𝑐<0, then
2 real solutions one real solution no real solutions
2 intercepts one intercept no intercepts

8. Exs. 3/4 cont. Determine number of real solutions and x-intercepts of:
𝑥 2 +8𝑥−3=0
𝑎=1, 𝑏=8, 𝑐=−3
8 2 −4 1 −3
64+12 76
Two solutions and two intercepts
Steps
1. get = to 0
2. use discriminant
𝑏 2 −4𝑎𝑐
3. simplify

9. Exs. 3/4 cont. Determine number of solutions and x-intercepts of:
9𝑥 2 +1=6𝑥
−6𝑥 −6𝑥
9𝑥 2 −6𝑥+1=0
𝑎=9, 𝑏=−6, 𝑐=1
(−6) 2 −4 9 1
36−36
One solution and one x-int
Determine number or solutions and x-intercepts of:
𝑦= 2𝑥 2 +3𝑥+9
𝑎=2, 𝑏=3, 𝑐=9
3 2 −4(2)(9)
9−72
-68
No solution and no x-int

10. Ex. 5 Choosing Any Method Method When to Use Factoring
Easily factored, 𝑥 2 +𝑏𝑥+𝑐
Square Roots
𝑥 2 −𝑑=0
Complete the Square
𝑎𝑥 2 +𝑏𝑥+𝑐 when a is one and b is even
Quadratic Formula
Anytime! Always Works!!
𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎