This video deals with the factorisation of x3 expression. Through this video, you will discover that maths is after-all not that difficult to handle.
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Thursday, 24 July 2008
Quadratic Formula | Common Application Mistakes
Solving Quadratic equation using formula is one of the easiest method and is well-liked by many students.
It is a straight-forward method where we can plug in numbers directly into the quadratic formula to get the answers to the equation.
A general form of the quadratic equation is ax2 + bx + c = 0.
The quadratic formula is x = [-b +- sqrt(b2-4ac) ]/ 2a
where x is the answers to the quadratic equation.
Example: 3x2 + 4x - 1 = 0 is a quadratic equation.
To solve the above, what we need is to identify the value for "a", "b" and c" ( called the coefficient).
Here a = 3, b= 4 and c = -1.
Next putting the numbers of above into the quadratic formula will allow us to get the answers directly. That is it! Nothing difficult!
However mistakes using quadratic formula do occur !
Why?
Never see the coefficient properly - that's why!
To apply the quadratic formula, we need to clearly identify the correct coefficient.
Example : 5x + 6x2 - 4 = 0.
Here a = 6 (not 5) since "a" belongs to the x2 term in the quadratic equation.
and b = 5 (not 6) as "b" belongs to the x term of the equation.
c = -4 (no doubt about it)
Message: You do not look for the position of the a, b and c. You look for the term associated with the x2 and x symbol.
Thus to use quadratic formula to solve quadratic equation, the only caution is for you to identify the correct coefficient.
Another common that is always made :
The general quadratic equation is ax2 + bx + c = 0.
NOTE: the equation = 0 .
If the quadratic question is 4x2 - 3x + 2 = x, what then are the value of 'a, 'b' and c'?
We need to ensure that the given equation matches the general quadratic equation form before we can proceed to identify the 'a', 'b' and 'c'.
The answer for above is 'a' = 4, 'b' = -3 -1 = -4, and 'c' = 2.
Advice: Just be careful that the final quadratic equation must = 0 before we do anything.
Clear?
Maths needs some form of mental discipline to get results. It serves you good in the long run.
.
It is a straight-forward method where we can plug in numbers directly into the quadratic formula to get the answers to the equation.
A general form of the quadratic equation is ax2 + bx + c = 0.
The quadratic formula is x = [-b +- sqrt(b2-4ac) ]/ 2a
where x is the answers to the quadratic equation.
Example: 3x2 + 4x - 1 = 0 is a quadratic equation.
To solve the above, what we need is to identify the value for "a", "b" and c" ( called the coefficient).
Here a = 3, b= 4 and c = -1.
Next putting the numbers of above into the quadratic formula will allow us to get the answers directly. That is it! Nothing difficult!
However mistakes using quadratic formula do occur !
Why?
Never see the coefficient properly - that's why!
To apply the quadratic formula, we need to clearly identify the correct coefficient.
Example : 5x + 6x2 - 4 = 0.
Here a = 6 (not 5) since "a" belongs to the x2 term in the quadratic equation.
and b = 5 (not 6) as "b" belongs to the x term of the equation.
c = -4 (no doubt about it)
Message: You do not look for the position of the a, b and c. You look for the term associated with the x2 and x symbol.
- 'a' always belongs t the term with the power of 2 (e.g. x2),
- 'b' belongs to the term with power of 1 (e.g. x), and
- 'c' belongs to the term with power of 0.
Thus to use quadratic formula to solve quadratic equation, the only caution is for you to identify the correct coefficient.
Another common that is always made :
The general quadratic equation is ax2 + bx + c = 0.
NOTE: the equation = 0 .
If the quadratic question is 4x2 - 3x + 2 = x, what then are the value of 'a, 'b' and c'?
We need to ensure that the given equation matches the general quadratic equation form before we can proceed to identify the 'a', 'b' and 'c'.
The answer for above is 'a' = 4, 'b' = -3 -1 = -4, and 'c' = 2.
Advice: Just be careful that the final quadratic equation must = 0 before we do anything.
Clear?
Maths needs some form of mental discipline to get results. It serves you good in the long run.
.
Ways to Solve Mathematical Word Problems
What is Mathematical Word Problem?
You may have encountered maths questions given in words instead of mathematical equations or expressions.
You may also come across situational questions or problems given as real-life scenario for solving using maths as a tool.
This word problems describes a certain situation with a problem within them.
The goal of the word problem is for you to gather relevant information in order to solve the problem.
What then is the appropriate strategy you need to use for solving this type of word problem?
Here I suggest the below steps:-
Step 1:
You have to seek out the objectives of the question.
Step 2:
From the identified objectives, list out the information given and required.
Step 3:
Relate the listed information to the objectives for the purpose of checking their relevance.
How do you know whether they are relevant?
Recall any maths equations or formulae having the given maths information as their variables within. If they exist within the maths equations, then they are relevant.
Otherwise they are given as extra and can be ignored since they are there to confuse the unaware.
This crucial step is the part that tests for true understanding!
Step 4:
Using the relevant information, compute for result or conclusion.
Word problem is a special way of questioning. It calls for a unique way to handle maths.
It enables you, the student, to use any acquired maths knowledge to solve issues described in words.
Keeping the goal in mind, and solving the problems through proper step-by-step approach trains you in problem-solving and thinking skill which comes in handy in real situations.
Therefore mathematical word problem is a useful method that teaches maths with a long term impact on a person's intellectual development.
Interesting isn't it that maths has such great importance!
.
You may have encountered maths questions given in words instead of mathematical equations or expressions.
You may also come across situational questions or problems given as real-life scenario for solving using maths as a tool.
This word problems describes a certain situation with a problem within them.
The goal of the word problem is for you to gather relevant information in order to solve the problem.
What then is the appropriate strategy you need to use for solving this type of word problem?
Here I suggest the below steps:-
Step 1:
You have to seek out the objectives of the question.
Step 2:
From the identified objectives, list out the information given and required.
Step 3:
Relate the listed information to the objectives for the purpose of checking their relevance.
How do you know whether they are relevant?
Recall any maths equations or formulae having the given maths information as their variables within. If they exist within the maths equations, then they are relevant.
Otherwise they are given as extra and can be ignored since they are there to confuse the unaware.
This crucial step is the part that tests for true understanding!
Step 4:
Using the relevant information, compute for result or conclusion.
Word problem is a special way of questioning. It calls for a unique way to handle maths.
It enables you, the student, to use any acquired maths knowledge to solve issues described in words.
Keeping the goal in mind, and solving the problems through proper step-by-step approach trains you in problem-solving and thinking skill which comes in handy in real situations.
Therefore mathematical word problem is a useful method that teaches maths with a long term impact on a person's intellectual development.
Interesting isn't it that maths has such great importance!
.
Wednesday, 23 July 2008
Order of Precedence for Maths
In our daily life, we encounter many rules. The rules are in place to keep things in order.
Similarly in maths, we also do have rules to keep the mathematical computations in proper order.
This post is created for the sake of some learners of maths who even at high school or upper secondary level still has not master this.
The first important rule before learning maths, I believe , is to master the Order of Precedence.
What is this Order of Precedence?
It is the priority level of the elementary mathematical operators, +, -, / , x and ().
They are used in a certain pre-determined order.
Let's see some maths examples.
Example1: 3 + 4 x 2
Example2: (3 + 4) x 2
Example3: 6 / 2 + 3
Example4: 12 / (3 - 1)
Example5: 4 - 2 + 3 x 2
Order of Precedence:
1st priority of use: Brackets ( )
2nd Priority of use: Multiply "x" or Divide "/"
3rd priority of use: Add "+" or Subtract "-"
Let us analyse the above maths equations.
Example1: 3 + 4 x 2
Base on the Order of Precedence, we need to perform the maths operation "x" first.Why?
We need to know the exact meaning of the maths operator "x".
In a x 3, we mean it to be a + a + a , that is, to add "a" 3 times.
Therefore Example1 can be re-written 3 + 4 + 4 since ONLY 4 is x 2, which is 4 + 4.
If we do addition "+" first, followed by x 2, Example1 becomes 3 + 4 + 3 + 4 (which is wrong!).
The correct answer: 3 + 4 x 2 = 3 + 8 = 11
NOTE: If we desire to have 3 + 4 added twice, we then need the maths operator bracket ().
(3 + 4) x 2 ==> 3 + 4 + 3 + 4 .
The brackets isolate the (3 + 4) from the multiply operator.
This is the result for Example2 also.
Example3 is 6 / 2 + 3 and base on the Order of Precedence, we need to perform the maths operation Divide "/" first before the Addition operation "+".
6 / 2 = 3 (the sub-working) which makes Example3 become 3 + 3 = 6.
NOTE: Example3 does not have ( ). Therefore 6 / 2 is done first due to "/" having higher priority.
If Example3 is modified to 6 / (2 + 3), the maths equation has the result = 6 / 5 since ( ) gets done first.
This explains the result for Example4 is 12 / (3 - 1) = 12 / 2 = 6.
How about Example5? 4 - 2 + 3 x 2
Again the maths operation "x' has to be performed first ==> 3 x 2 = 3 + 3 = 6
Example5 becomes 4 - 2 + 3 x 2 = 4 -2 + 6 = 2 + 6 = 8 (answer).
Tip: Practice with the Order of Precedence in mind.
Through sincere constant practice, the concept and meaning of these maths operators will be understood and retained.
Only through strengthening this importance knowledge, you can then avoid many unnecessary maths errors,and move forward doing maths with ease.
:)
.
Similarly in maths, we also do have rules to keep the mathematical computations in proper order.
This post is created for the sake of some learners of maths who even at high school or upper secondary level still has not master this.
The first important rule before learning maths, I believe , is to master the Order of Precedence.
What is this Order of Precedence?
It is the priority level of the elementary mathematical operators, +, -, / , x and ().
They are used in a certain pre-determined order.
Let's see some maths examples.
Example1: 3 + 4 x 2
Example2: (3 + 4) x 2
Example3: 6 / 2 + 3
Example4: 12 / (3 - 1)
Example5: 4 - 2 + 3 x 2
Order of Precedence:
1st priority of use: Brackets ( )
2nd Priority of use: Multiply "x" or Divide "/"
3rd priority of use: Add "+" or Subtract "-"
Let us analyse the above maths equations.
Example1: 3 + 4 x 2
Base on the Order of Precedence, we need to perform the maths operation "x" first.Why?
We need to know the exact meaning of the maths operator "x".
In a x 3, we mean it to be a + a + a , that is, to add "a" 3 times.
Therefore Example1 can be re-written 3 + 4 + 4 since ONLY 4 is x 2, which is 4 + 4.
If we do addition "+" first, followed by x 2, Example1 becomes 3 + 4 + 3 + 4 (which is wrong!).
The correct answer: 3 + 4 x 2 = 3 + 8 = 11
NOTE: If we desire to have 3 + 4 added twice, we then need the maths operator bracket ().
(3 + 4) x 2 ==> 3 + 4 + 3 + 4 .
The brackets isolate the (3 + 4) from the multiply operator.
This is the result for Example2 also.
Example3 is 6 / 2 + 3 and base on the Order of Precedence, we need to perform the maths operation Divide "/" first before the Addition operation "+".
6 / 2 = 3 (the sub-working) which makes Example3 become 3 + 3 = 6.
NOTE: Example3 does not have ( ). Therefore 6 / 2 is done first due to "/" having higher priority.
If Example3 is modified to 6 / (2 + 3), the maths equation has the result = 6 / 5 since ( ) gets done first.
This explains the result for Example4 is 12 / (3 - 1) = 12 / 2 = 6.
How about Example5? 4 - 2 + 3 x 2
Again the maths operation "x' has to be performed first ==> 3 x 2 = 3 + 3 = 6
Example5 becomes 4 - 2 + 3 x 2 = 4 -2 + 6 = 2 + 6 = 8 (answer).
Tip: Practice with the Order of Precedence in mind.
Through sincere constant practice, the concept and meaning of these maths operators will be understood and retained.
Only through strengthening this importance knowledge, you can then avoid many unnecessary maths errors,and move forward doing maths with ease.
:)
.
How To Solve Exponential Equations
Exponential equations? What are they?
They are simply equations written in the indices form xn = Y.
By looking at their simple form, we can deduce that their solution will not deviate much from simplicity also.
Lets look at a few examples.
Ex 1: Solve 2x = 16
We change the 16 to 24. This allows us to compare using the mathematical logic that when the base(2) is the same, the power of the base must be the same. 2x = 24.
Therefore x = 4 (Answer).
NOTE: How about doing something like 2x = 15? This case we need Logarithm !
Ex 2: Solving the exponential equation 22k - 3(2k) + 2 = 0.
This is slightly challenging in that the exponential equation is of the quadratic form.
Maths Tips: Let the 2k be u. And 22k be u2. This simplifies the outlook of the exponential equation without changing its meaning.
The new modified question becomes u2 - 3u +2 = 0. Is this simpler to solve? Must be!
Moving on....
Using factorisation, we get (u - 1)(u -2) = 0.
Which means (u - 1) = 0 or (u - 2) = 0.
Therefore u = 1 or u = 2. Replacing the u = 2k back to u,
2k = 1= 20 or 2k = 2
Answer: k = 0 or k = 1.
From the above examples, they showed that with proper planning and understanding, solving exponential equations can be simple and fun.
Therefore strive to understand the principles of the topics, and all will be fine. :)
.
They are simply equations written in the indices form xn = Y.
By looking at their simple form, we can deduce that their solution will not deviate much from simplicity also.
Lets look at a few examples.
Ex 1: Solve 2x = 16
We change the 16 to 24. This allows us to compare using the mathematical logic that when the base(2) is the same, the power of the base must be the same. 2x = 24.
Therefore x = 4 (Answer).
NOTE: How about doing something like 2x = 15? This case we need Logarithm !
Ex 2: Solving the exponential equation 22k - 3(2k) + 2 = 0.
This is slightly challenging in that the exponential equation is of the quadratic form.
Maths Tips: Let the 2k be u. And 22k be u2. This simplifies the outlook of the exponential equation without changing its meaning.
The new modified question becomes u2 - 3u +2 = 0. Is this simpler to solve? Must be!
Moving on....
Using factorisation, we get (u - 1)(u -2) = 0.
Which means (u - 1) = 0 or (u - 2) = 0.
Therefore u = 1 or u = 2. Replacing the u = 2k back to u,
2k = 1= 20 or 2k = 2
Answer: k = 0 or k = 1.
From the above examples, they showed that with proper planning and understanding, solving exponential equations can be simple and fun.
Therefore strive to understand the principles of the topics, and all will be fine. :)
.
Indices | Commonly Made Mistakes
Humans are imperfect. Therefore to make errors in maths computation, we are reminded that we are human!
Does that mean we can excuse ourselves for making maths errors?
No...... There are maths basics that we cannot afford to make errors.
Therefore to avoid making errors in Indices, read the explanation below.
Who knows, by reading this post, you may end up one day as a professor!
Therefore to cut short everything, here it goes ......
*******************************************************************************
The format for indices are one of the simplest in maths. In its simplest form ax = Y.
But mistakes are still being made! Why?
Firstly, the writing skill of students has to improve.
When the power of a number (base) has to be written higher up and smaller than the base number, it has to be so.
Error: ax = Y written as ax = Y ==> the power "x" has changed meaning.
Secondly, understanding of the Laws of Indices is not fully digested.
Error: a0 = 1 misunderstood as a0 = 0 ==> concept of power is wrong as it became multiplication instead of division (e.g. a2 / a2 = 1).
Thirdly, the function of brackets is not fully used.
Brackets can isolate the powers when many manipulations or operations are done to simplify index terms.
Error-1: x4 / x 2-n = x4-(2-n) = x2+n became = x4-2-n = x2-n . The sign for the small "n" is wrong.
Error-2: (xK)2 = x2 K2 taken as x K2 forgetting that "x" is affected by the power 2 also.
How to solve this common indices mistakes?
Practice and practice, and clarify thoughts. This is different from learning other subjects where practicing of questions are not that intensive. Maths is a flexible and though-provoking subject that embodies many thinking skills. So Work hard and Enjoy Maths!
:-)
.
Does that mean we can excuse ourselves for making maths errors?
No...... There are maths basics that we cannot afford to make errors.
Therefore to avoid making errors in Indices, read the explanation below.
Who knows, by reading this post, you may end up one day as a professor!
Therefore to cut short everything, here it goes ......
*******************************************************************************
The format for indices are one of the simplest in maths. In its simplest form ax = Y.
But mistakes are still being made! Why?
Firstly, the writing skill of students has to improve.
When the power of a number (base) has to be written higher up and smaller than the base number, it has to be so.
Error: ax = Y written as ax = Y ==> the power "x" has changed meaning.
Secondly, understanding of the Laws of Indices is not fully digested.
Error: a0 = 1 misunderstood as a0 = 0 ==> concept of power is wrong as it became multiplication instead of division (e.g. a2 / a2 = 1).
Thirdly, the function of brackets is not fully used.
Brackets can isolate the powers when many manipulations or operations are done to simplify index terms.
Error-1: x4 / x 2-n = x4-(2-n) = x2+n became = x4-2-n = x2-n . The sign for the small "n" is wrong.
Error-2: (xK)2 = x2 K2 taken as x K2 forgetting that "x" is affected by the power 2 also.
How to solve this common indices mistakes?
Practice and practice, and clarify thoughts. This is different from learning other subjects where practicing of questions are not that intensive. Maths is a flexible and though-provoking subject that embodies many thinking skills. So Work hard and Enjoy Maths!
:-)
.
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