1. Section 9.5 Day 1 Solving Quadratic Equations by using the Quadratic Formula
Algebra 1

2. Learning Targets
Solve quadratic equations by using the Quadratic Formula
Define discriminant
Use the discriminant to determine the number of solutions of a quadratic equation
Summarize the differences between solving a quadratic equation by graphing, factoring, using square roots, completing the square, and the quadratic formula.

3. 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 Quadratic Formula
For 𝑎 𝑥 2 +𝑏𝑥+𝑐=0, the solutions are
𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎

4. Solving using Quadratic Equations - Example 1
Solve 𝑥 2 −12𝑥=−20 using the Quadratic Formula
1. 𝑥 2 −12𝑥+20=0: 𝑎=1, 𝑏=−12, 𝑐=20
2. Plug in: 𝑥= − −12 ± −12 2 −
3. Simplify: 𝑥= 12± 144−80 2 = 12± = 12±8 2
𝑥= =10 and 𝑥= 12−8 2 =2
𝑥=10 𝑎𝑛𝑑 𝑥=2

5. Solving using Quadratic Equations - Example 2
Solve 10 𝑥 2 −5𝑥=25 by using the Quadratic Formula
1. 10 𝑥 2 −5𝑥−25=0, 𝑎=10, 𝑏=−5, 𝑐=−25
2. 𝑥= − −5 ± −5 2 −4 10 − = 5± = 5±
𝑥= 5± = 1±

6. Solving using Quadratic Equations - Example 3
Solve 6 𝑥 2 −2𝑥+1=0
1. 𝑎=6, 𝑏=−2, 𝑐=1
2. Plug in: 𝑥= − −2 ± −2 2 −
3. Simplify: 𝑥= 2± 4−24 4 = 2± −20 4
4. No Solutions

7. Solving using Quadratic Equations - Example 4
Solve 9 𝑥 2 +12𝑥=−4
1. 9 𝑥 2 +12𝑥+4=0;𝑎=9, 𝑏=12, 𝑐=4
2. Plug in: 𝑥= − 12 ± −
3. Simplify: 𝑥= −12± 144− = −12± =− 12 18
4. 𝑥=− =− 2 3

8. Discriminant
The discriminant of the quadratic formula is the expression under the radical.
𝑏 2 −4𝑎𝑐
It is used to determine the number of solutions a quadratic equation will have

9. Discriminant It comes from 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2 𝑎
If 𝑏 2 −4𝑎𝑐>0, there are 2 solutions
If 𝑏 2 −4𝑎𝑐=0, there is 1 solution
If 𝑏 2 −4𝑎𝑐<0, there are no real solutions

10. Determine the number of solutions Example 1
How many solutions does 𝑥 2 +2𝑥+5=0 produce?
𝑎=1, 𝑏=2, 𝑐=5
𝑏 2 −4𝑎𝑐= − =4−20=−16<0
No real solutions

11. Determine the number of solutions Example 2
How many solutions does 𝑥 2 +10𝑥+25=0 produce?
𝑎=1, 𝑏=10, 𝑐=25
𝑏 2 −4𝑎𝑐= − =100−100=0
One real solution

12. Determine the number of solutions Example 3
How many solutions does 2 𝑥 2 −7𝑥+2=0 produce?
𝑎=2, 𝑏=−7, 𝑐=2
𝑏 2 −4𝑎𝑐= −7 2 − =49−16=33>0
Two Real Solutions

13. Summary Factoring Graphing Using Square Roots Completing the Square
GCF, “X” Method, Difference of Squares
Zero Product Property
Graphing
Standard, Vertex, Intercept
Roots, Zeros, X-Intercepts
Using Square Roots
Taking the square root of each side
Completing the Square
𝑏 2𝑎 2 , taking the square root of each side
Quadratic Formula
𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎