The document defines quadratic equations as polynomial equations of the second degree where the highest exponent on the variable is 2. It provides the general form of a quadratic equation as ax2 + bx + c = 0 where a, b, and c are constants and a ≠ 0. Roots or solutions of a quadratic equation are the values that make the equation equal to 0. The document discusses several methods for solving quadratic equations including factoring, using the quadratic formula, and completing the square. It provides examples of solving quadratic equations using each of these methods.
This will help you on how to solve quadratic equations by factoring.
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This will help you on how to solve quadratic equations by factoring.
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Mathematics 9 Lesson 1-B: Solving Quadratic Equations using Quadratic FormulaJuan Miguel Palero
This powerpoint presentation discusses or talks about the topic or lesson Solving Quadratic Equations using the Quadratic Formula. It also discusses the steps in solving quadratic equations using the method of Quadratic Formula.
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2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
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In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
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.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
2. Definition
• In mathematics, a quadratic equation is a
polynomial equation of the second degree. The
general form is
2
ax bx c 0
• where x represents a variable or an unknown,
and a, b, and c are constants with a ≠ 0. (If
a = 0, the equation is a linear equation.)
• The constants a, b, and c are called
respectively, the quadratic coefficient, the linear
coefficient and the constant term or free term.
3. Quadratic & Roots
Quadratic: A polynomial of degree=2
y= ax2+bx+c
is a quadratic equation. (a 0)
Here is an example of one:
• The name Quadratic comes from "quad" meaning square,
because the variable gets squared (like x2).
• It is also called an "Equation of Degree 2" (because of the "2"
on the x)
4. Roots
A real number α is called a root of the
quadratic equation ,a≠0
if aα2 + bα2 + c = 0.
If α is a root of ,then we
say that:
(i) x= α satisfies the equation ax2+bx+c =0
Or (ii) x= α is a solution of the equation
ax2+bx+c =0
The Root of a quadratic equation ax2+bx+c
=0 are called zeros of the polynomial
ax2+bx+c .
5. More Examples of Quadratic
Equations
In this one a=2, b=5 and c=3.
This one is a little more tricky:
Where is a? In fact a=1, as we don't usually
write "1x2“ b = -3 and where is c? Well, c=0, so
is not shown.
Oops! This one is not a
quadratic equation, because it is missing x2 (in
other words a=0, and that means it can't be
quadratic)
6. Hidden Quadratic Equations!
So far we have seen the "Standard Form" of a Quadratic
Equation:
But sometimes a quadratic equation doesn't look like that..!
Here are some examples of different form:
In disguise In Standard Form a, b and c
x2 = 3x -1 Move all terms to x2 - 3x + 1 = 0 a=1, b=-3, c=1
left hand side
2(w2 - 2w) = 5 Expand (undo the 2w2 - 4w - 5 = 0 a=2, b=-4, c=-5
brackets), and
move 5 to left
z(z-1) = 3 Expand, and move a=1, b=-1, c=-3
3 to left
z2 - z - 3 = 0
5 + 1/x - 1/x2 = 0 Multiply by x2 5x2 + x - 1 = 0 a=5, b=1, c=-1
7. How To Solve It?
There are 3 ways to find the solutions:
We can Factor the Quadratic (find what to multiply to make the
Quadratic Equation)
We can Complete the Square, or
We can use the special Quadratic Formula:
Thus ax2+bx+c =0 has two roots α and β, given by
b b 2 4ac b b 2 4 ac
α= β=
2a 2a
8. Discriminant
The expression b2 - 4ac in the formula
It is called the Discriminant, because it can "discriminate"
between the possible types of answer.It can be denoted by “D”
when b2 - 4ac, D is positive, you get two real solutions
when it is zero you get just ONE real solution (both answers are
the same)
when it is negative you get two Complex solutions
Value of D Nature of Roots Roots
D>0 Real and Unequal [(-b±√D)/2a]
D=0 Real and Equal Each root = (-b/2a)
D<0 No real roots None
9. Using the Quadratic Formula
Just put the values of a, b and c into the Quadratic Formula, and do
the calculation
Example: Solve 5x² + 6x + 1 = 0
Coefficients are: a = 5, b = 6, c = 1
Quadratic Formula: x = [ -b ± √(b2-4ac) ] / 2a
Put in a, b and c:
6 62 4 5 1
x= 2 5
6 36 20
Solve: x =
10
6 16
x=
10
6 4
x=
10
x = -0.2 or -1
11. Factoring Quadratics
To "Factor" (or "Factorize") a Quadratic is to find what to multiply
to get the Quadratic
It is called "Factoring" because you find the factors (a factor is
something you multiply by)
Example
The factors of x2 + 3x - 4 are:
(x+4) and (x-1)
Why? Well, let us multiply them to see:
(x+4)(x-1)
= x(x-1) + 4(x-1)
= x2 - x + 4x - 4
= x2 + 3x – 4
• Multiplying (x+4)(x-1) together is called Expanding.
• In fact, Expanding and Factoring are opposites:
12. Examples of Factor
To solve by factoring:
1. Set the equation equal to zero.
2. Factor. The factors will be linear expressions.
3. Set each linear factor equal to zero.
4. Solve both linear equations.
Example: Solve by factoring x2 + 3x = 0
x2 + 3x = 0 set equation to zero
x( x + 3) = 0 factor
x=0 , x+3=0
x = -3 set the linear factors equal to zero and
solve the linear equation
13. Completing the Square
Solving General Quadratic Equations by Completing
the Square:
"Completing the Square" is where we take a Quadratic Equation :
ax2 + bx + c = 0 and turn into a(x+d)2 + e = 0
We can use that idea to solve a Quadratic Equation (find where it
is equal to zero).
But a general Quadratic Equation can have a coefficient of a in
front of x2:
But that is easy to deal with ... just divide the whole equation by "a"
first, then carry on.
14. Steps
Now we can solve Quadratic Equations in 5 steps:
Step 1 Divide all terms by a (the coefficient of x2).
Step 2 Move the number term (c/a) to the right side of the
equation.
Step 3 Complete the square on the left side of the equation and
balance this by adding the same value to the right side of the
equation.
Step 4 Take the square root on both sides of the equation.
Step 5 Add or subtract the number that remains on the left side
of the equation to find x.
15. Example
Example 1: Solve x2 + 4x + 1 = 0
Step 1 can be skipped in this example since the coefficient of x2 is
1
Step 2 Move the number term to the right side of the equation:
x2 + 4x = -1
Step 3 Complete the square on the left side of the equation and
balance this by adding the same number to the right side of the
equation:
x2 + 4x + 4 = -1 + 4
(x + 2)2 = 3
Step 4 Take the square root on both sides of the equation:
x + 2 = ±√3 = ±1.73 (to 2 decimals)
Step 5 Subtract 2 from both sides:
x = ±1.73 – 2 = -3.73 or -0.27