1.5 Quadratic Equations.pdf

1.5 Quadratic Equations
Chapter 1 Equations and Inequalities

Concepts and Objectives
⚫ Objectives for this section:
⚫ Review factoring quadratic expressions
⚫ Review solving quadratic equations by factoring

Factoring Polynomials
⚫ The process of finding polynomials whose product
equals a given polynomial is called factoring.
⚫ For example, since 4x + 12 = 4(x + 3), both 4 and x + 3 are
called factors of 4x + 12.
⚫ A polynomial that cannot be written as a product of two
polynomials of lower degree is a prime polynomial.
⚫ One nice aspect of this process is that it has a built-in
check: whatever factors you come up with, you should
be able to multiply them and get your starting
expression.

Factoring Out the GCF
Factor out the greatest common factor from each
polynomial:
⚫ GCF: y2
⚫ GCF: 2t
⚫
GCF: 7m + 1
5 2
9y y
+
2
6 8 12
x t xt t
+ +
( ) ( ) ( )
3 2
14 1 28 1 7 1
m m m
+ − + − +
( )
3
2
9 1
y y +
( )
2
6
2 3 4
x
t x
+ +
( ) ( ) ( )
2
4
7 1 2 1 1 1
m m
m  
+ −
+ − +
 

Factoring Out the GCF (cont.)
We can clean up that last problem just a little more:
( ) ( ) ( )
( ) ( ) ( )
( )
( )( )
 
+ + − + −
 
 
+ + + − + −
 
 
+ + + − − −
 
+ −
2
2
2
2
7 1 2 1 4 1 1
7 1 2 2 1 4 1 1
7 1 2 4 2 4 4 1
7 1 2 3
m m m
m m m m
m m m m
m m

Factoring Trinomials
If you have an expression of the form ax2 +bx + c, you can
use one of the following methods to factor it:
⚫ X-method (a = 1): If a = 1, this is the simplest method to
use. Find two numbers that multiply to c and add up
to b. These two numbers will create your factors.
⚫ Example: Factor x2 ‒ 5x ‒ 14.
‒14
‒7 2
‒5
( )( )
2
5 14 7 2
x x x x
− − = − +
c
b

Factoring Trinomials (cont.)
⚫ Reverse box: If a is greater than 1, you can use the
previous X method to split the middle term (ac goes on
top) and use either grouping or the box method, and
then find the GCF of each column and row.
⚫ Example: Factor
Now, find the GCF of each line.
− −
2
4 5 6
y y
‒24
‒8 3
‒5
y ‒2
4y 4y2 ‒8y
3 3y ‒6
ac
b
( )( )
− − = + −
2
4 5 6 4 3 2
y y y y

⚫ Grouping: This method is about the same as the Reverse
Box, except that it is not in a graphic format.
⚫ Example: Factor 2
2 6
x x
− −
‒12
‒4 3
‒1
( ) ( )
( ) ( )
( )( )
2 2
2
2 6 2 6
2 4 3 6
2 2 3 2
2 2 3
4 3
x x x
x x x
x x
x
x
x
x x
− − = −
= − + −
=
=
+
−
−
− + −
+

⚫ I prefer using a method called the Mustang method:
This method is named after the mnemonic “My Father
Drives A Red Mustang”, where the letters stand for:
⚫ If you are solving an equation, you don’t have to bother
moving the denominators; you can just stop at “R”.
M Multiply a and c.
F Find factors using the X method. Set up ( ).
DA Divide the numeric terms by a if necessary.
R Reduce any fractions.
M Move any denominators to the front of the variable.

⚫ Example: Factor 2
5 7 6
x x
+ −
M Multiply (5)(‒6) = ‒30
F Find factors:
DA Divide by a
R Reduce fractions
M Move the denominator
‒30
10 ‒3
7
( )( )
10 3
x x
+ −
10 3
5 5
x x
  
+ −
  
  
( )
3
2
5
x x
 
+ −
 
 
( )( )
2 5 3
x x
+ −

Sidebar: Calculator Shortcut
⚫ If you have a TI-83/84, one way your calculator can help
you find the factors is to do the following:
⚫ In o, set Y1= to ac/X (whatever a and c are)

⚫ In Y2=, go to ½; then select , À, and À. This
should put Y1 in the Y2= line. Then enter Ã„.

⚫ Go to the table (ys). What you’re looking for is
a Y2 that equals b. The values of X and Y1 are your
two factors.

⚫ To do the same thing in Desmos:
⚫ Go to desmos.com/calculator (or use the link I’ve
added to the left-hand side of Canvas).
⚫ On the first line, type f (x) = ac/x (again, use a and c
from your equation)
⚫ On the next line, type g(x) = f (x)+x
⚫ On the third line, type y = b (whatever your b value is)
⚫ Look on the graph for the two points where the
horizontal line crosses g(x). The x values represent
your two factors.

⚫ Example:
(I had to enter the point values to get them to show up on
the screen capture – you don’t have to do that.)

Factoring Binomials
⚫ If you are asked to factor a binomial (2 terms), check
first for common factors, then check to see if it fits one of
the following patterns:
⚫ Note: There is no factoring pattern for a sum of
squares (a2 + b2) in the real number system.
Difference of Squares a2 ‒ b2 = a + ba ‒ b
Sum/Diff. of Cubes ( )( )
3 3 2 2
a b a b a ab b
 =  +

Factoring Binomials (cont.)
Examples
⚫ Factor
⚫ Factor
⚫ Factor
2
4 81
x −
3
27
x −
3
3 24
x +
( )
( )( )
2 2
2 9
2 9 2 9
x
x x
= −
= − +
( )( )
3 3
2
3
3 3 9
x
x x x
=
+
−
= − +
( ) ( )
( )( )
3 3 3
2
3 8 3 2
3 2 2 4
x x
x x x
= + = +
= + − +

Quadratic Equations
⚫ A quadratic equation is an equation that can be written
in the form
where a, b, and c are real numbers and 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

Factoring Quadratic Equations
⚫ Solving by 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.

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
 
−
 
 

Factoring Binomials (cont.)
EVERY TIME YOU DO THIS:
A KITTEN DIES
( )
2 2 2
x y x y
+ = +
Remember:
Hat tip: https://mathcurmudgeon.blogspot.com/2014/01/do-this-and-bunny-dies.html

For Next Class
⚫ Section 1.5 homework (MyMathLab)
⚫ Quiz 1.5 (Canvas)
⚫ Optional – read section 1.6 in your text
Reminder: You may retake either of these as many times
as you like until Sunday at 11:59 pm.

1.5 Quadratic Equations.pdf

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1.5 Quadratic Equations.pdf