This document discusses methods for solving quadratic and cubic equations. It begins by introducing quadratic equations in standard form and methods for solving them, including factoring, completing the square, and using the quadratic formula. It then discusses properties related to the square root and applies them to solving quadratic equations. The document concludes by introducing cubic equations that are the sum or difference of cubes, and provides an example of solving one using factoring.

1. 1.4 Quadratic Equations
Chapter 1 Equations and Inequalities

2. Concepts and Objectives
⚫ Quadratic Equations
⚫ Solve quadratic equations, finding all solutions
⚫ Cubic Equations
⚫ Solve the sum or difference of two cubes

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
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 {±k}
⚫ 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

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 this method 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 a cb 4, ,2 1== =−
( ) ( ) ( )( )
( )
x
2
2
2
1 1 4
2
4−  −
=
− −
 −  −
= =
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. Classwork
⚫ College Algebra
⚫ Page 119: 14-30, page 109: 38-54, page 89: 44-54
(skip 48) (all even)