This document provides instruction on factoring quadratic equations. It begins by reviewing factoring polynomials and trinomials. It then discusses factoring binomials using difference of squares, sum/difference of cubes, and other patterns. Finally, it explains that a quadratic equation can be solved by factoring if it can be written as a product of two linear factors. An example demonstrates factoring a quadratic equation by finding the two values that make each factor equal to zero.
* Factor the greatest common factor of a polynomial.
* Factor a trinomial.
* Factor by grouping.
* Factor a perfect square trinomial.
* Factor a difference of squares.
* Factor the sum and difference of cubes.
* Factor expressions using fractional or negative exponents.
* Solve polynomial equations by factoring
* Solve equations with radicals and check the solutions
* Solve equations with rational exponents
* Solve equations that are quadratic in form
* Solve absolute value equations
* 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.
* Identify intervals on which a function increases, decreases, or is constant
* Use graphs to locate relative maxima or minima
* Test for symmetry
* Identify even or odd functions and recognize their symmetries
* Understand and use piecewise functions
* Factor the greatest common factor of a polynomial.
* Factor a trinomial.
* Factor by grouping.
* Factor a perfect square trinomial.
* Factor a difference of squares.
* Factor the sum and difference of cubes.
* Factor expressions using fractional or negative exponents.
* Solve polynomial equations by factoring
* Solve equations with radicals and check the solutions
* Solve equations with rational exponents
* Solve equations that are quadratic in form
* Solve absolute value equations
* 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.
* Identify intervals on which a function increases, decreases, or is constant
* Use graphs to locate relative maxima or minima
* Test for symmetry
* Identify even or odd functions and recognize their symmetries
* Understand and use piecewise functions
* Solve equations in one variable algebraically.
* Solve a rational equation.
* Find a linear equation.
* Given the equations of two lines, determine whether their graphs are parallel or perpendicular.
* Write the equation of a line parallel or perpendicular to a given line.
* Solve equations involving rational exponents
* Solve equations using factoring
* Solve equations with radicals and check the solutions
* Solve absolute value equations
* Solve other types of equations
* Recognize characteristics of graphs of polynomial functions.
* Use factoring to find zeros of polynomial functions.
* Identify zeros and their multiplicities.
* Determine end behavior.
* Understand the relationship between degree and turning points.
* Graph polynomial functions.
* Use the Intermediate Value Theorem.
* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
* Express an exponential model in base e
* Construct perpendicular and angle bisectors
* Use bisectors to solve problems
* Identify the circumcenter and incenter of a triangle
* Use triangle segments to solve problems
* Solve equations in one variable algebraically.
* Solve a rational equation.
* Find a linear equation.
* Given the equations of two lines, determine whether their graphs are parallel or perpendicular.
* Write the equation of a line parallel or perpendicular to a given line.
* Solve equations involving rational exponents
* Solve equations using factoring
* Solve equations with radicals and check the solutions
* Solve absolute value equations
* Solve other types of equations
* Recognize characteristics of graphs of polynomial functions.
* Use factoring to find zeros of polynomial functions.
* Identify zeros and their multiplicities.
* Determine end behavior.
* Understand the relationship between degree and turning points.
* Graph polynomial functions.
* Use the Intermediate Value Theorem.
* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
* Express an exponential model in base e
* Construct perpendicular and angle bisectors
* Use bisectors to solve problems
* Identify the circumcenter and incenter of a triangle
* Use triangle segments to solve problems
* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
* Find the distance between two points
* Find the midpoint of two given points
* Find the coordinates of an endpoint given one endpoint and a midpoint
* Find the coordinates of a point a fractional distance from one end of a segment
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Introduce functions and function notation
* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
* Recognize graphs of common functions.
* Graph functions using vertical and horizontal shifts.
* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
* Combine transformations.
* Determine whether a relation or an equation represents a function.
* Evaluate a function.
* Use the vertical line test to identify functions.
* Identify the domain and range of a function from its graph
* Identify intercepts from a function’s graph
* Solve counting problems using the Addition Principle.
* Solve counting problems using the Multiplication Principle.
* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
* Find the number of subsets of a given set.
* Solve counting problems using permutations involving n non-distinct objects.
* Use summation notation.
* Use the formula for the sum of the first n terms of an arithmetic series.
* Use the formula for the sum of the first n terms of a geometric series.
* Use the formula for the sum of an infinite geometric series.
* Solve annuity problems.
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
* Find the common difference for an arithmetic sequence.
* Write terms of an arithmetic sequence.
* Use a recursive formula for an arithmetic sequence.
* Use an explicit formula for an arithmetic sequence.
* Write the terms of a sequence defined by an explicit formula.
* Write the terms of a sequence defined by a recursive formula.
* Use factorial notation.
* Identify characteristics of each type of conic section
* Identify a conic section from its equation in general form
* Identifying the eccentricities of each type of conic section
* Graph parabolas with vertices at the origin.
* Write equations of parabolas in standard form.
* Graph parabolas with vertices not at the origin.
* Solve applied problems involving parabolas.
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
2. Concepts and Objectives
⚫ Objectives for this section:
⚫ Review factoring quadratic equations
⚫ Review solving quadratic equations by factoring
3. 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.
4. Factoring Out the GCF
Factor out the greatest common factor from each
polynomial:
⚫
⚫
⚫
5 2
9y y
+
2
6 8 12
x t xt t
+ +
( ) ( ) ( )
3 2
14 1 28 1 7 1
m m m
+ − + − +
5. Factoring Out the GCF
Factor out the greatest common factor from each
polynomial:
⚫ GCF: y2
⚫ GCF: 2t
⚫
GCF: 7m + 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
+ −
+ − +
6. 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
7. 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
8. 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
4y2 ‒8y
3y ‒6
ac
b
9. 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 − −
2
4 5 6
y y
‒24
‒8 3
5
y ‒2
4y 4y2 ‒8y
3 3y ‒6
( )( )
− − = + −
2
4 5 6 4 3 2
y y y y
10. Factoring Trinomials (cont.)
⚫ 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
− − = −
= − + −
=
=
+
−
−
− + −
+
11. Factoring Trinomials (cont.)
⚫ My preferred method is 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 factors by a if necessary.
R Reduce any fractions.
M Move any denominators to the front of the variable.
12. Factoring Trinomials (cont.)
⚫ 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
+ −
13. 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)
14. Sidebar: Calculator Shortcut
⚫ In Y2=, go to ½; then select , À, and À. This
should put Y1 in the Y2= line. Then enter Ä.
15. Sidebar: Calculator Shortcut
⚫ 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.
16. Sidebar: Calculator Shortcut
⚫ 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.
17. Sidebar: Calculator Shortcut
⚫ 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.)
18. 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 + ba ‒ b
Sum/Diff. of Cubes ( )( )
3 3 2 2
a b a b a ab b
= +
19. 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
= + = +
= + − +
20. 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
21. 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
22. 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.
24. 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
−
25. 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.