1.4 Quadratic Equations
Learning Objectives:
 Solve quadratic equations by factoring.
 Solve quadratic equations by isolating the variable.
 Solve quadratic equations by using the quadratic formula.
 Use the discriminant to interpret what types of solutions exist.
Definition of Quadratic & the Square Root Property
Quadratic-______________________________________________________________
________________________________________________________________________
________________________________________________________________________
Example: _____________________________________________
Non-examples: _________________________________________
_________________________________________
_________________________________________
Square Root Property:
Example:
𝑥2 = 7
______________________
______________________
What does it mean if you have √7? _____________________________________
__________________________________________________________________
__________________________________________________________________
In general: ________________________________________________________
Factoring & Isolating the Variable
Example 1:
𝑥 2 + 6𝑥 = 0
___________________________
Factor
____________ or ____________
Zero-Product Property
____________ or ____________
Solve for x in each
1
The solution set is ________________________.
Example 2:
2𝑥 2 = 𝑥 + 3
___________________________
Put into standard form
___________________________
Factor (FOIL to check __________)
___________ or _____________
Zero-Product Property
___________ or _____________
Solve for x in each
The solution set is ________________________.
Example 3:
3𝑥 2 − 6 = 0
___________________________
Add _____ to both sides
___________________________
Divide both sides by ___
___________________________
Take the square root of both sides
The solution set is ________________________.
Finding the Quadratic Formula I
Solve: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 (𝑎 ≠ 0)
_________________________________________ Divide each side by ______
_________________________________________ Subtract ______ from each side
𝑏 1
𝑏
_________________________________________ Complete the Square (𝑎 ∙ 2 = 2𝑎)
_________________________________________ Take the square root of both sides
_________________________________________ Subtract _______ from each side
_________________________________________ Combine the fractions / Simplify
Finding the Quadratic Formula II
2
Find the zeros / Find the x-intercepts / Find the roots
𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑎≠0
___________________________________ Set equal to zero
___________________________________ Subtract _____ from each side
___________________________________ Divide each side by ________
___________________________________ Complete the Square
___________________________________ Simplify left side
___________________________________ Take the square root of both sides
___________________________________ Subtract ________ from each side
___________________________________ Simplify
Determining Solutions: ____________________________________________________
_____________________________________________________
_____________________________________________________
Using the Quadratic Formula
Recall the standard form of a Quadratic Equation: _______________________________
Now, recall the Quadratic Formula: ___________________________________________
Example: Solve
25
2
𝑥 2 − 30𝑥 + 18 = 0.
_________________________________
______________________________
3
_________________________________
______________________________
𝑎 = _________
𝑏 = _________
𝑐 = _________
Plug into the Quadratic Formula: _______________________________________
Simplify:
_____________________________________________________
_____________________________________________________
The solution set is: _______________
The Discriminant’s Role in the Quadratic Formula I
Remember a quadratic equation ______________________________________________
Remember the quadratic formula _____________________________________________
Example 1:
2𝑥 2 + 4𝑥 + 1 = 0
Calculate the Discriminant:
____________________________________
____________________________________
If the discriminant is ______________ that means we have ____________
___________________________________________________________.
Using the Quadratic Formula: ___________________________________
____________________________________
____________________________________
Example 2:
2𝑥 2 + 4𝑥 + 2 = 0
Calculate the Discriminant:
____________________________________
If the discriminant is ______________ that means we have ____________
___________________________________________________________.
4
Using the Quadratic Formula: ___________________________________
____________________________________
Example 3:
2𝑥 2 + 4𝑥 + 3
Calculate the Discriminant:
____________________________________
If the discriminant is ______________ that means we have ____________
___________________________________________________________.
Using the Quadratic Formula: ___________________________________
____________________________________
The Discriminant’s Role in the Quadratic Formula II
Recall the Quadratic Formula: _______________________________________________
The discriminant is: _______________________________________________________
Discriminant of a Quadratic Equation
For a quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0:
1) _______________________________________________________________
2) _______________________________________________________________
3) _______________________________________________________________
Example: 3𝑥 2 − 5𝑥 + 1 = 0.
𝑎 = _________
𝑏=
_________
𝑐 = _________
Evaluate the discriminant: ____________________________________________
What does the discriminant tell us for this example? _______________________
Example: 3𝑥 2 + 2 = 4𝑥
Put in standard form: ________________________________________________
5
𝑎 = _________
𝑏=
_________
𝑐 = _________
Evaluate the discriminant: ____________________________________________
What does the discriminant tell us for this example? _______________________
Quadratic Equation at the Beach
Solve 10 = 3𝑥 2 − 2
_________________________________
Add ____ to each side
_________________________________
Divide each side by ______
_________________________________
Take the square root of both sides
The ANSWER is… _________________
6