The document explains the concept of quadratic equations, represented in standard form as ax² + bx + c = 0, and outlines a systematic approach for solving them through factoring. It discusses the zero-factor property, which states that if the product of two factors equals zero, then at least one of the factors must be zero. A step-by-step method is provided to factorize and solve quadratic equations, followed by examples illustrating the process.

1. Solving Equations by Factoring
Definition of Quadratic Equations
Zero-Factor Property
Strategy for Solving Quadratics

2. Standard Form Quadratic
Equation
Quadratic equations can be written in the form
ax2
+ bx + c = 0
where a, b, and c are real numbers with a  0.
Standard form for a quadratic equation
is in descending order equal to zero.
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3. Examples of Quadratic Equations
p
p 18
81
2


18
9
2

 x
x
25
2

y
0
81
18
2


 p
p
Standard Form
0
18
9
2


 x
x
0
25
2


y
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4. Zero-Factor Property
If a and b are real numbers
and if ab =0, then
a = 0 or
b = 0.
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5. Solve the equation (x + 2)(2x - 1)=0
• By the zero factor property we know...
• Since the product is equal to zero then one of the factors must be
zero.
0
)
2
( 

x
2


x
OR (2 1) 0
x  
1
2 
x
2
1
2
2

x 2
1

x
}
2
1
,
2
{

x
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6. Solve the equation. Check your answers.
0
)
2
)(
5
( 

 x
x
5

x
OR 0
2 

x
2


x { 2, 5}
x  
0
5 

x Solution Set
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7. Solve each equation. Check your answers.
0
)
3
5
( 

x
x
0

x OR
0
3
5 

x
5
3


x
}
0
,
5
3
{


x
0

x
Solution Set
3
5 

x
5
3
5
5 

x
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8. Solving a Quadratic Equation by Factoring
Step 1 Write the equation in standard
form.
Step 2 Factor completely.
Step 3 Use the zero-factor property.
Set each factor with a variable equal
to zero.
Step 4 Solve each equation produced
in step 3.
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9. Solve.
18
9
2

 x
x
0
18
9
2


 x
x
0
)
3
)(
6
( 

 x
x
}
3
,
6
{

x BACK

10. Solve.
0
7
2

 x
x
0
)
7
( 

x
x
}
7
,
0
{

x
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11. Number Of Solutions
•The degree of a polynomial
is equal to the number of
solutions.
x
x
x 3
2 2
3


Three solutions!!!

12. Example
x (x + 1)(x – 3) = 0
Set each of the three factors equal to 0.
x = 0 x + 1 = 0
x = -1
x – 3 = 0
x = 3
Solve the resulting equations.
Write the solution set.
x = {0, -1, 3}
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13. Solve the following equations.
1. x2
– 25 = 0
2. x2
+ 7x – 8 = 0
3. x2
– 12x + 36 = 0
4. c2
– 8c = 0
  
x x
  
5 5 0
  
x x
  
8 1 0
  
x x
  
6 6 0
c c
( )
 
8 0
 
5
,
5 
 
8
,
1 
 
6
 
8
,
0

14. 1. Get a value of zero on one side of the
equation.
2. Factor the polynomial if possible.
3. Apply the zero product property by
setting each factor equal to zero.
4. Solve for the variable.