This document outlines methods to solve quadratic equations, including factoring, the square root property, completing the square, and using the quadratic formula. It explains the significance of the discriminant in determining the nature of solutions and provides examples for each method. Objectives include understanding how to implement these techniques and their applications in problem-solving.

1. 2.5 Quadratic Equations
Chapter 2 Equations and Inequalities

2. Concepts and Objectives
⚫ Objectives for this section:
⚫ Solve quadratic equations by factoring.
⚫ Solve quadratic equations by the square root
property.
⚫ Solve quadratic equations by completing the square.
⚫ Solve quadratic equations by using the quadratic
formula.

3. Quadratic Equations
⚫ A quadratic equation is an equation that can be written
in the form
where a, b, and c are real numbers, with a  0. This is
called standard form.
⚫ A quadratic equation can be solved by factoring,
graphing, completing the square, or by using the
quadratic formula.
⚫ Graphing and factoring don’t always work, but
completing the square and the quadratic formula will
always provide the solution(s).
+ + =
2
0
ax bx c

4. Factoring Quadratic Equations
⚫ Factoring works because of the zero-factor property:
⚫ If a and b are complex numbers with ab = 0, then
a = 0 or b = 0 or both.
⚫ To solve a quadratic equation by factoring:
⚫ Put the equation into standard form (= 0).
⚫ If the equation has a GCF, factor it out.
⚫ Using the method of your choice, factor the quadratic
expression.
⚫ Set each factor equal to zero and solve both factors.

5. Factoring Quadratic Equations
Example: Solve by factoring.
− − =
2
2 15 0
x x

6. Factoring Quadratic Equations
Example: Solve by factoring.
The solution set is
− − =
2
2 15 0
x x
= = − = −
2, 1, 15
a b c –30
–1
–6 5
6 5
0
2 2
x x
  
− + =
  
  
( )
5
3 0
2
x x
 
− + =
 
 
5
3 0 or 0
2
x x
− = + =
= −
5
, 3
2
x
5
, 3
2
 
−
 
 

7. Square Root Property
⚫ If x2 = k, then
⚫ Both solutions are real if k > 0 and often written as
⚫ Both solutions are imaginary if k < 0, and written as
⚫ If k = 0, there is only one distinct solution, 0.
or
x k x k
= = −
 
i k

 
k


8. Square Root Property (cont.)
Example: What is the solution set?
⚫ x2 = 17
⚫ x2 = ‒25
⚫ ( )
2
4 12
x − =

9. Square Root Property (cont.)
Example: What is the solution set?
⚫ x2 = 17
⚫ x2 = ‒25
⚫ ( )
2
4 12
x − =
 
17

 
5i

4 12
4 2 3
x
x
− = 
= 
25 5
x i
=  − = 
 
4 2 3

17
x = 
Remember to simplify
any radicals!

10. Completing the Square
⚫ As the last example shows, we can use the square root
property if x is part of a binomial square.
⚫ It is possible to manipulate the equation to produce a
binomial square on one side and a constant on the other.
We can then use the square root property to solve the
equation. This method is called completing the square.

11. Completing the Square (cont.)
Solving a quadratic equation (ax2 + bx + c = 0) by
completing the square:
⚫ If a  1, divide everything on both sides by a.
⚫ Isolate the constant (c) on the right side of the equation.
⚫ Add ½b2 to both sides.
⚫ Factor the now-perfect square on the left side.
⚫ Use the square root property to complete the solution.

12. Completing the Square (a = 1)
Example: Solve x2 ‒ 4x ‒ 14 = 0 by completing the square.

13. Completing the Square (a = 1)
Example: Solve x2 ‒ 4x ‒ 14 = 0 by completing the square.
( )
2
2
2 2 2
2
4 14 0
4 14
4 14
1
2
2 2
2 18
2
2 8
3
x x
x x
x x
x
x
x
− − =
− =
+ = +
=
− = 
−

−
=

14. Completing the Square (a  1)
Example: Solve 4x2 + 6x + 5 = 0 by completing the square.

15. Completing the Square (a  1)
Example: Solve 4x2 + 6x + 5 = 0 by completing the square.
2
2
2
2
2 2
2
2
4 6 5 0
3 5
0 Divide by 4
2 4
3 5
2 4
3 3 5 3 1 3
Add to each side
2 4 4 4 2 2
3 11
4 16
3 11
4 16
3 11
4 4
x x
x x
x x
x x
x
x
x i
+ + =
+ + =
+ = −
 
     
+ + = − +
     
 
     
 
 
+ = −
 
 
+ =  −
= − 
3 11
The solution set is
4 4
i
 
 
− 
 
 
 

16. Quadratic Formula
⚫ The solutions of the quadratic equation ,
where a  0, are
⚫ Example: Solve
+ + =
2
0
ax bx c
−  −
=
2
4
2
b b ac
x
a
= −
2
2 4
x x

17. Quadratic Formula
⚫ Example: Solve = −
2
2 4
x x
− + =
2
2 4 0
x x a c
b 4
, ,
2 1
=
= =
−

18. Quadratic Formula
⚫ Example: Solve
The solution set is
= −
2
2 4
x x
− + =
2
2 4 0
x x −
= =
= 4
, ,
1
2
a c
b
( ) ( ) ( )( )
( )
−  −
− −
=
2
1 1
2
4 2 4
2
x
 −  −
= =
1 1 32 1 31
4 4

=
1 31
4
i  
 

 
 
 
1 31
4 4
i

19. The Discriminant
⚫ The discriminant is the quantity under the radical in the
quadratic formula: b2 − 4ac.
⚫ When the numbers a, b, and c are integers, the value of
the discriminant can be used to determine whether the
solutions of a quadratic equation are rational, irrational,
or nonreal complex numbers.
− 
=
−
2
2
4
b
x
b ac
a
Discriminant

20. The Discriminant (cont.)
⚫ The number and type of solutions based on the value of
the discriminant are shown in the following table.
⚫ Remember, a, b, and c must be integers.
Discriminant Number of Solutions Type of Solution
Positive, perfect
square
Two (can be factored) Rational
Positive, not a
perfect square
Two Irrational
Zero One (a double solution) Rational
Negative Two Nonreal complex

21. Classwork
⚫ College Algebra 2e
⚫ 2.5: 6-18 (even); 2.4: 6-16 (even); 2.3: 28, 30, 32, 33,
36, 37, 44, 46-48
⚫ 2.5 Classwork Check
⚫ Quiz 2.4