* 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.

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