The document defines quadratic equations and provides methods for solving them, including factoring, using the square root property, completing the square, and the quadratic formula. It also discusses the discriminant and how it determines the number and type of solutions. Examples are provided to illustrate each solving method step-by-step. Key aspects covered include the definition and forms of quadratic equations, the zero-product principle for factoring, and how the discriminant indicates whether solutions are real, repeated, or imaginary.

1. DEFINITION OF A QUADRATIC EQUATION
A quadratic equation in x is an equation that can be written in
the general form
ax bx c2 0,
where a, b, and c are real numbers, with a ≠ 0.
A quadratic equation in x is also called a second-degree
polynomial equation in x.
THE ZERO-PRODUCT PRINCIPLE
To solve a quadratic equation by factoring, we apply the zero-
product principle which states that:
If the product of two algebraic expressions is zero, then at least
one of the factors is equal to zero.
If AB = 0, then A = 0 or B = 0.
SOLVING A QUADRATIC EQUATION BY FACTORING
2. Factor completely.

2. 3. Apply the zero-product principle, setting each factor
containing a variable equal to zero.
4. Solve the equations in step 3.
5. Check the solutions in the original equation.
EXAMPLE: SOLVING QUADRATIC
EQUATIONS BY FACTORING (1 OF 3)
Solve by factoring: 2x2 x 1.
Step 1 Move all nonzero terms to one side and obtain zero on
the other side.
2x x2 1 0
Step 2 Factor
(2x− 1)(x + 1) = 0
EQUATIONS BY FACTORING (2 OF 3)
Steps 3 and 4 Set each factor equal to zero and solve the
resulting equations.
(2x− 1)(x + 1) = 0

3. EXAMPLE: SOLVING QUADRATIC
EQUATIONS BY FACTORING (3 OF 3)
Step 5 Check the solutions in the original equation.
2x x2 1
Check
EXAMPLE: SOLVING QUADRATIC
EXAMPLE: SOLVING QUADRATIC EQUATIONS BY
FACTORING
•
Solution

4. • Step 1- Move all nonzero terms to one side and obtain zero on
the other.
• Step 2 Factor- 2x(2x-1)=0
• Step 3 and 4- Set each factor equal to zero and solve the
resulting equations.
2x=0 or 2x-1=0 x=0
2x=1
x=1/2
• Step 5
B) SOLUTION

5. CHECK POINT SOLVE BY FACTORING
SOLVING QUADRATIC EQUATIONS BY THE SQUARE
ROOT PROPERTY

6. EXAMPLE: SOLVING QUADRATIC EQUATIONS BY THE
SQUARE ROOT PROPERTY
Solve by the square root property:
SOLUTION
SOLUTION

7. SOLUTION
CHECK POINT: SOLVE BY THE SQUARE ROOT
PROPERTY:
COMPLETING THE SQUARE
EXAMPLE: CREATING PERFECT SQUARE TRINOMIALS
BY COMPLETING THE SQUARE

9. CHECK POINT: SOLVING A QUADRATIC EQUATION BY
COMPLETING THE SQUARE
EXAMPLE: SOLVING A QUADRATIC EQUATION BY
COMPLETING THE SQUARE

10. CHECK POINT: SOLVE BY COMPLETING THE SQUARE.
THE QUADRATIC FORMULA

11. EXAMPLE: SOLVING A QUADRATIC EQUATION USING
THE QUADRATIC
FORMULA (1 OF 2)
Solve using the quadratic formula:
2x2 2x 1 0
a = 2, b = 2, c = −1
EXAMPLE: SOLVING A QUADRATIC EQUATION USING
THE QUADRATIC FORMULA (2 OF 2)
CHECK POINT:

12. THE DISCRIMINANT
The discriminant of the quadratic equation determines the
number and type of solutions.
THE DISCRIMINANT AND THE KINDS OF SOLUTIONS TO
A X SQUARED + B X + C = 0
• If the discriminant is positive, there will be two unequal real
solutions.
• If the discriminant is zero, there is one real (repeated)
solution.
• If the discriminant is negative, there are two imaginary

13. solutions.
EXAMPLE: USING THE DISCRIMINANT
CORNELL NOTES
SHEET
Name: ________John Doe_________________________

14. Class: ___MAC1105___________ Topic: _Complex
Numbers___
Date: ____01___/ _04___/ __2022____
Period _Spring 2022___
QUESTIONSNOTES
The imaginary unit is denoted by the letter “i”, and i = sqrt ( -1)
Addition: (3 + 3i) + (2 – i) = 3 + 3i + 2 – i = 5 – 2i
Subtraction: (4 + 3i) – (-2 – 5i) = 4 + 3i + 2 + 5i = 6 + 8i
Why were called imaginary?
Multiplication: 4( 3 + 2i) = 4(3) + 4(2i) = 12 + 8i
Who introduced the complex
Complex conjugate of a + bi is a - bi

15. numbers in Math?
Division: 3/(2 + i) = 3(2 -i)/(2+i)(2-i) = (6 – 3i)/ (4 +1) = (6 –
3i)/5 =
= 6/5 – 3i/5
How are complex numbers
used in real life?
Powers of i: i^ 21. Divide 21 by 4, the remainder 1 is the new
exponent, so
I^21 = i^1 = i

17. SUMMARY: Write 4 or more sentences describing specific
learning from these notes.
_____________________________________________________
_____________________________________
__I learned what a complex number is; the different operations
that we can perform: addition, subtraction, _
__multiplication, division, and powers of i.
_______________________________________________
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