This document summarizes key concepts about quadratic equations, including: - Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula. - Completing the square involves manipulating the equation into a perfect square trinomial form. - The quadratic formula provides the solutions to any quadratic equation in standard form. - Cubic equations that are the sum or difference of cubes can be factored and solved. - Literal quadratic equations can be solved for a specified variable using techniques like the square root property or quadratic formula. - The discriminant determines whether the solutions to a quadratic equation are rational, irrational, or complex numbers.

1. 1.4 Quadratic Equations
Chapter 1 Equations and Inequalities

2. Concepts and Objectives
⚫ Quadratic Equations
⚫ Solve quadratic equations, finding all solutions by
⚫ Factoring
⚫ Completing the square
⚫ Quadratic formula
⚫ Using the discriminant
⚫ Cubic Equations
⚫ Solve the sum or difference of two cubes
⚫ Solving a literal quadratic equation for a variable

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
0ax 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 0x x

6. Factoring Quadratic Equations
Example: Solve by factoring.
The solution set is
− − =2
2 15 0x x
= = − = −2, 1, 15a b c –30
–1
–6 5− + − =2
2 6 5 15 0x x x
( ) ( )− + − =2 3 5 3 0x x x
( )( )+ − =2 5 3 0x x
+ = − =2 5 0 or 3 0x 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.
orx k x k= = −
 i k
 k

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

9. Square Root Property (cont.)
Example: What is the solution set?
⚫ x2 = 17
⚫ x2 = ‒25
⚫ ( )
2
4 12x − =
 17
 5i
4 12
4 2 3
x
x
− = 
= 
25 5x i=  − = 
 4 2 3
17x = 
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
22 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
0ax bx c
−  −
=
2
4
2
b b ac
x
a
= −2
2 4x x

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

18. Quadratic Formula
⚫ Example: Solve
The solution set is
= −2
2 4x x
− + =2
2 4 0x x −= == 4, ,12a cb
( ) ( ) ( )( )
( )
−  −− −
=
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. Cubic Equations
⚫ We will mainly be working with cubic equations that are
the sum or difference of two cubes:
a3  b3 = 0
⚫ Equations of this form factor as
⚫ To solve this, set each factor equal to zero and solve.
(Use the Quadratic Formula or Completing the Square
for the quadratic factor.)
( )( ) + =2 2
0a b a ab b

20. Cubic Equations
Example: Solve 8x3 + 125 = 0

21. Cubic Equations
Example: Solve 8x3 + 125 = 0 ( ) + =
3 3
2 5 0x
( )( )+ − + =2
2 5 4 10 25 0x x x

22. Cubic Equations
Example: Solve 8x3 + 125 = 0 ( ) + =
3 3
2 5 0x
( )( )+ − + =2
2 5 4 10 25 0x x x
+ = − + =2
2 5 0 or 4 10 25 0x x x

23. Cubic Equations
Example: Solve 8x3 + 125 = 0 ( ) + =
3 3
2 5 0x
( )( )+ − + =2
2 5 4 10 25 0x x x
+ = − + =2
2 5 0 or 4 10 25 0x x x
+ =
= −
= −
2 5 0
2 5
5
2
x
x
x

24. Cubic Equations
Example: Solve 8x3 + 125 = 0 ( ) + =
3 3
2 5 0x
( )( )+ − + =2
2 5 4 10 25 0x x x
+ = − + =2
2 5 0 or 4 10 25 0x x x
+ =
= −
= −
2 5 0
2 5
5
2
x
x
x
( ) ( )( )
( )
 − −
=
2
10 10 4 4 25
2 4
x
 −  
= = =
10 300 10 10 3 5 5 3
8 8 4
i i

25. Cubic Equations
Example: Solve 8x3 + 125 = 0 ( ) + =
3 3
2 5 0x
( )( )+ − + =2
2 5 4 10 25 0x x x
+ = − + =2
2 5 0 or 4 10 25 0x x x
+ =
= −
= −
2 5 0
2 5
5
2
x
x
x
( ) ( )( )
( )
 − −
=
2
10 10 4 4 25
2 4
x
 −  
= = =
10 300 10 10 3 5 5 3
8 8 4
i i
  
−  
  
5 5 5 3
,
2 4 4
i
The solution set is

26. Literal Quadratic Equations
⚫ To solve a quadratic equation for a specified variable in
a formula or in a literal equation, we usually apply the
square root property (taking the square root of both
sides) or the quadratic formula.
⚫ Examples: Solve for the specified variable. Use  when
taking square roots.
a) b)
2
, for
4
d
A d

= ( )2
0 , forrt st k r t− = 

27. Literal Quadratic Equations (cont.)
(a)
2
4
A
d
=
2
4A d= 
24A
d=

4A
d = 

2 A
d
 
= 
 
2 A
d
 
=

Note: In this equation, these
variables don’t necessarily have any
concepts attached to them, so we are
taking both the positive and negative
square roots. If the variables were to
represent something like distance or
area, we would consider only the
positive square root.

28. Literal Quadratic Equations (cont.)
(b) 2
rt s kt− =
2
0rt st k− − = a = r, b = −s, c = −k
( ) ( ) ( )( )
( )
2
4
2
r
r
t
ks s−− − −−
=
2
4
2
s s rk
r
 +
=

29. 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
4b
x
b ac
a
Discriminant

30. 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

31. Classwork
⚫ 1.4 Assignment
⚫ Page 119: 14-30 (even); page 109: 40-68 (4);
page 100: 36-44 (even)
⚫ 1.4 Classwork Check
⚫ Quiz 1.3