I warmly welcome every steemian in week 4 course about algebra!
It's important to discuss introduction of algebra and important terminologies about algebra before getting involved into in depth of main topic that I am teaching you guys today!
Algebra is basically a major branch of mathematics which consists of study of variables and their relationships with eachother.If I talk about algebra then It includes usage of different symbols, equations, and formulas for solving problems and for modeling real world applications.
Linear equations
If I talk about linear equation then its an equation where highest power of the variable is 1.
Algebraic example can be written as 2x + 3 = 5
Practical Example:
Suppose there is a bakery selling 250 loaves of bread each day. If they are making profit of $0.50 for each loaf and their per day total profit is $125. If they are trying for increasing their profit to $150 then how much loaves they need for selling ?Suppose that here x are number of additional loaves sold out.
Quadratic equations
If I talk about quadratic equation then its an equation where highest power of the variable is 2.
Algebraic example is x^2 + 4x + 4 = 0
Practical example:
Suppose a ball is throwing in upward direction from ground with its initial velocity of 20 meter per second.Suppose height of ball above ground is given by following equation h(t) = -4.9t^2 + 20t where h is height in meters and t is representing time in seconds. What is maximum height reached by that ball?
If you have to solve linear equations by different methods then you can use addition, subtraction, multiplication and division methods which can be more clear for you by following these different examples for each method;
Addition Method
In this method you have for adding same value to both sides for isolating variable.
Example: x - 3 = 7
Now you need for adding 3 at both sides: x = 10
Subtraction Method
For implementing subtraction method you need of subtracting same value from both sides for isolating variable.
Example: x + 2 = 9
Now there's a need of subtracting 2 from both sides: x = 7
Multiplication Method
If I talk about multiplication method then you need of multiplying both sides by same values for isolating variables.
Example: 2x = 12
Here is a need of multiplying both sides with 1/2: x = 6
Division Method
If we have to implement division method then you need for dividing both sides with same value for isolating variable.
Example: 4x = 20
At last there's a need of dividing both sides with 4: x = 5
If you have to solve for quadratic equations then you can solve them by use of factoring, quadratic formula and graphing method!
Factoring Method
- Factoring of quadratic equation in two binomials.
- Setting of each binomial equal to 0 and solving for x.
Take an example: x^2 + 5x + 6 = 0
Factoring: (x + 2)(x + 3) = 0
Solving: x + 2 = 0 or x + 3 = 0
x = -2 or x = -3
Quadratic Formula Method
- By using quadratic formula which is x = (-b ± √(b^2 - 4ac)) / 2a
- Plugging in values of a, b, c.
Take an example as x^2 + 4x + 4 = 0
Suppose that;
a = 1
b = 4
c = 4
x = (-(4) ± √((4)^2 - 4(1)(4))) / 2(1)
x = (-4 ± √(16 - 16)) / 2
x = -2
Graphing Method
- You need for graphing quadratic equation at coordinate plane.
- Now there's a need of finding x-intercepts, which is used for representing solutions.
Take an example as x^2 + 2x - 3 = 0
Graph will be presented as crossing x-axis at x = -3 and x = 1.
If I talk about methods for solving linear and quadratic equations then you can use following methods in both cases;
Linear Equations
Quadratic Equations
Addition Method
In this method you have for adding same value to both sides for isolating variable.
Example: x - 3 = 7
Now you need for adding 3 at both sides: x = 10
Subtraction Method
For implementing subtraction method you need of subtracting same value from both sides for isolating variable.
Example: x + 2 = 9
Now there's a need of subtracting 2 from both sides: x = 7
Multiplication Method
If I talk about multiplication method then you need of multiplying both sides by same values for isolating variables.
Example: 2x = 12
Here is a need of multiplying both sides with 1/2: x = 6
Division Method
If we have to implement division method then you need for dividing both sides with same value for isolating variable.
Example: 4x = 20
At last there's a need of dividing both sides with 4: x = 5
Factoring Method
- Factoring of quadratic equation in two binomials.
- Setting of each binomial equal to 0 and solving for x.
Take an example: x^2 + 5x + 6 = 0
Factoring: (x + 2)(x + 3) = 0
Solving: x + 2 = 0 or x + 3 = 0
x = -2 or x = -3
Quadratic Formula Method
- By using quadratic formula which is x = (-b ± √(b^2 - 4ac)) / 2a
- Plugging in values of a, b, c.
Take an example as x^2 + 4x + 4 = 0
Suppose that;
a = 1
b = 4
c = 4
x = (-(4) ± √((4)^2 - 4(1)(4))) / 2(1)
x = (-4 ± √(16 - 16)) / 2
x = -2
Graphing Method
- You need for graphing quadratic equation at coordinate plane.
- Now there's a need of finding x-intercepts, which is used for representing solutions.
Take an example as x^2 + 2x - 3 = 0
Graph will be presented as crossing x-axis at x = -3 and x = 1
If I talk about linear and quadratic equations then they have multiple applications in real life.So have a look at some applications below;
Linear equation real life applications
• If there is a company producing widgets at a cost of $5 for each unit and there's a fixed cost of $100. If they are producing x units then their total cost is 5x + 100 so I'm cost calculation it has applications.
• If there's a car travelling at constant speed of 60 miles for each hour then suppose If it is travelling for x hours then total distance it is covering is 60x miles so here is also real life application of linear equation.
Quadratic equation real life applications
• If I talk about trajectory of a projectile like for a ball or for a missile then it may be modeled by the use of quadratic equations.
• If I talk about another application of quadratic equation then these are useful in optimizing functions like for reduction of cost for production or maximising of area of a rectangle.
These examples are useful for illustrating that how linear and quadratic equations are helpful in applying in real-world problems as well as these are essential tools for solution of problems of different fields.
| • | Task 1 |
|---|
• Explain difference between linear and quadratic equations. Provide examples of each type of system of equation and describe their general forms.
| • | Task 2 |
|---|
• Describe methods for solving quadratic equations and share pros and cons for each method.
| • | Task 3 |
|---|
• Solve for linear equation 3x + 2 = 11 and show value of x?
• Solve for this quadratic equation x^2 + 2x - 6 = 0.
(You are required to solve these problems at paper and then share clear photographs for adding a touch of your creativity and personal effort which should be marked with your username)
| • | Task 4 |
|---|
Scenario number 1
Suppose Ali have $15 for spending at snacks. He is buying a pack of chips for $3. How much money he have left?
Suppose x is amount of money Ali has left.
Equation: x + 3 = 15
Share a solution for x
(Solve the above scenerio based questions and share step by step that how you reach to your final outcome)
Scenario number 2
Suppose there's a ball which is thrown in upward direction from ground with initial velocity of 20 m/s and height of ball above ground is presented by following equation;
h(t) = -5t^2 + 20t
Here h is height in meters and t is time in seconds.
Share about maximum height reached by this ball!
Please solve for h!
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Contest deadline is from 6th Janauary to 12th Janauary.
Thank you for participating in s22 w4 learning challange!
| Criteria | Status |
|---|---|
| Plagiarism 🆓 | ✅ |
| AI 🆓 | ✅ |
| Category | Remark |
|---|---|
| Task 1 | /1.5 |
| Task 2 | /1.5 |
| Task 3 | /3 |
| Task 4 | /3 |
| Post quality | 1 |
| Total grade | /10 |
Teacher's Observation
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Recommendations or suggestions for improvement of content
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