1. Quadratic Equations Quadratic Formula:
A Quadratic Equation is an equation where there
is only one variable and the highest power is 2.
The Standard form is given by:
Quadratic Formula:
Types of Solutions:
Discriminant: The Radical Part of the Quadratic Formula:
two unequal real Solutions
one repeated Solution:
Multiplicity of two
Two imaginary Solutions

2. Solving Quadratic Equations
(In General)
Note: these solutions can be combined
using the following notation:
To solve a Quadratic Equation in one variable:
1- List any restrictions on the Domain
2- Use Appropriate factoring methods, and set each factor
equal to zero.
3- If the Polynomial does not factor Complete the Square or
use the Quadratic Formula.

3. Solving Quadratic Equations
(Factoring Methods)
Can the equation be factored using one (ore more) of the following formulas, if so factor it and set each factor equal to zero to solve.
Look at the Number of Terms you have:
Two terms:
Sum of two Squares-
a2+b2 = PRIME – Cannot be factored
Difference of two squares
a2-b2 = (a-b)(a+b)
Sum of two Cubes
a3+b3 = (a+b)(a2-ab+b2)
Difference of two Cubes
a3-b3 = (a-b)(a2+ab+b2)
Three Terms:
General Form: ax2+bx+c
when a=1; (C Method)
C>0 (x+y1)(x+y2) or (x-y1)(x-y2)
where y1 and y2 are factors of c
C<0 (x+y1)(x-y2)
y1 and y2 are factors of c and
the larger factor gets the sign of b
when a1 Grouping- (AC Method);
1- Multiply A and C
2- Write factors that will multiply to get AC (watch the signs)
3- Add together the factors and circle the one that results in B.
4- Group two terms together using Parenthesis and factor out common factors of each parenthesis.
5- Write the repeated factor once, and the common factors together for your final
answer.
Perfect Square Trinomials
a2-2ab+b2 = (a-b)(a-b) = (a-b)2
a2+2ab+b2 = (a+b)(a+b) = (a+b)2
NOTE: Only works if A and C are perfect squares and
the middle coefficient is twice their product (2ac) -regardless of sign.
Four or more Terms:
Cubed Binomials
(a+b)3 = a3 + 3a2b + 3ab2 + b3
(a-b)3 = a3 – 3a2 b+ 3ab2 – b3

4. Solving Quadratic Equations (Completing the Square)
Method 1:
1- Set the equation equal to zero
2- Move the Constant c to the Right side
3- Divide all terms by the leading coefficient a
(if a is not already equal to 1)
4- Divide b by two and square that number
5- Add the result of step 4 to both sides of the
equation
6- Write the left side as a perfect square Binomial
7- Square root both sides of the equation
8- Get the variable by itself by addition or
subtraction (from the left side)
9- Check your answers
Note: Remember you should have two solutions
from the square root

5. Solving Quadratic Equations (Completing the Square)
Method 1:
EXAMPLE:
Solve by Completing the Square

6. Solving Quadratic Equations (Completing the Square)
Method 2:
1- Set the equation equal to zero
2- Divide x2 and x terms by the leading coefficient a
(if a is not already equal to 1)
3- Place a Parenthesis around those two terms and
leave a space for the new constant.
4- Divide b by two and square that number
5- Add the result of step 4 in the space for the
new constant, and add a times that number to
the Right side of the equation.
6- Write the left side as a perfect square Binomial
(with a as the leading coefficient)
7- Move the constant to the left side of equation
by addition or subtraction
8- Divide both sides by a
9- Square root both sides of the equation
10- Get variable alone on left side
(using addition or subtraction)
11- Check your answers

7. Solving Quadratic Equations (Completing the Square)
Method 2:
EXAMPLE:
Solve by Completing the Square

8. Solving Quadratic Equations
(Quadratic Formula)
EXAMPLE:
Solve by using the Quadratic Formula