QUADRATIC EQUATIONS : Completing The Square part.B (SPM, IGCSE, A-level...)

(Since my time is limited, so for every chapter I will skip some and only focus on parts which most of the students found problem with it. Maybe after I retire, then only I will complete those skipped parts.)


In previous chapter,
I mentioned that will go more in depth about Completing The Square.

What did I mean by that?

It is actually...........
.............
FORMULIZATION!

Maybe some of you will feel weird,
didn't I always against memorizing formula?

Well,
I am against memorizing formula WITHOUT UNDERSTANDING IT,
but I do not against memorizing the formula AFTER UNDERSTANDING IT.

If you read my previous two chapters about CTS and understood the principle behind it,
then it will be a lot easier to memorize the steps.

If you not yet read,
the links are here:
1st part: https://leonmathstuition.blogspot.com/2019/03/quadratic-equations-before-learning.html
2nd part: https://leonmathstuition.blogspot.com/2019/03/quadratic-equations-completing-square.html


Now,
I will use the example from previous chapter first:

Answer:

1)Make sure that no other number except 1 in front of x^2.

2)Move the number to the other side of equation.

3)Turn the left hand side into PSQ.(by dividing the number in front of x by 2, then square it.)

4)Factorise the PSQ on the left hand side.
5)Find x.

That's all.

The conclusion:
1)Make sure that no other number except 1 in front of x^2.
2)Move the number to the other side of equation.
3)Turn the left hand side into PSQ.(by dividing the number in front of x by 2, then square it.)
4)Factorise the PSQ on the left hand side.
5)Find x.

5 steps only,
quick & easy.

If there is anything that you don't really understand since you not yet read the previous two chapters,
the links are here:
1st part: https://leonmathstuition.blogspot.com/2019/03/quadratic-equations-before-learning.html
2nd part: https://leonmathstuition.blogspot.com/2019/03/quadratic-equations-completing-square.html

Last example:

Answer:

Next,
let's practice with some questions.

Since the steps of CTS are always repeating,
so just 5 questions enough:

Answer:

DONE.

Until here,
we are done with CTS,
but this is not the end yet.

At next chapter,
I will talk about the relationship between CTS and Formula Method.

Of course,
even if you understand all these,
it doesn't mean that you can get full marks in your exam.

UNDERSTANDING is just UNDERSTANDING,
SOLVING QUESTION is different.

At least if you understood,
it will help you a lot in solving questions.

Maths should be learnt easily,
Practice should be done accordingly,
exam should be faced seriously.

Thank you!

End.

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Related articles:
Before learning Completing The Square
Completing The Square part.A


Important Links:
CONTACT ME
ABOUT ME
MY MATHS BLOG(Chinese ver.)
MY YOUTUBE


Feel free to e-mail me if you have any problems:
leonleongmaths@gmail.com

QUADRATIC EQUATIONS : Completing The Square part.A (SPM, IGCSE, A-level...)

(Since my time is limited, so for every chapter I will skip some and only focus on parts which most of the students found problem with it. Maybe after I retire, then only I will complete those skipped parts.)

Today let's learn about CTS (Completing The Square).

At previous post,
I talked about Perfect Square Quadratic (PSQ),
since PSQ and CTS both relate to each other.

Why??

This is because before learning about CTS,

We need to be able to know what number should we fill in,
to make it become PSQ.

If you read my previous post,
you already know that the number is 16.

If you not yet read my previous post,
the link is here:
https://leonmathstuition.blogspot.com/2019/03/quadratic-equations-before-learning.html

Why 16??

It is very easy,
just take the -8x at the middle.

-8 divide by 2, get -4,
then square the -4, get 16.

After filling in 16, it became PSQ:

If you still don't really understand,
or you didn't read my previous post,
please don't be stubborn,
just read:
https://leonmathstuition.blogspot.com/2019/03/quadratic-equations-before-learning.html

Next,
let's practice some questions first.

For all the 10 questions below,
what number should we fill in to make it become PSQ:

Answer:

Okay,
let's start learning CTS now.

First let's have a look at an example question:

Answer:

We move the +3 to the other side of the equation,
then just do like the example above:

What number should we add to make it become PSQ?

Now we all know that the number is 16,
so:

Since it is an equation,
so when we +16 at the left hand side,
we need to +16 at the right hand side too.

For example: 2 + 3 = 5

If left hand side +4 : 2 + 3 + 4

Then will the right hand side still be 5??
OF COURSE NOT!

So both sides also +4.

2 + 3 + 4 = 5 + 4

Then only the left = right !

Let's get back to the example question.........

After +16,
the left hand side became PSQ,
then we can apply the Factorisation of PSQ that I mentioned in previous post:

Just like the square root of 9 can be +3 or -3,
so we need to consider that square root of 13 can be positive or negative too.

Of course,
you don't have to write until so troublesome when actually answering the question.

Let me show you the simplified version of answering:

One more example:

Answer:

That's all for now.

As usual,
10 questions to practice:

Answer:

We will go more in depth about CTS at next chapter.

Thank you!

End.

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Related articles:
Before learning Completing The Square


Important Links:
CONTACT ME
ABOUT ME
MY MATHS BLOG(Chinese ver.)
MY YOUTUBE


Feel free to e-mail me if you have any problems:
leonleongmaths@gmail.com

QUADRATIC EQUATIONS : Before learning about Completing The Square (SPM, IGCSE, A-LEVEL...)

(Since my time is limited, so for every chapter I will skip some and only focus on parts which most of the students found problem with it. Maybe after I retire, then only I will complete those skipped parts.)

From now on we will be learning about Completing The Square.

Huh??
Is there anything to learn??
Just remember the formula can already right??

Well.... yeah.
You can solve the question by memorizing the formula,
just like how you can solve Factorisation by using calculator.

HOWEVER..........

You will waste your chance to admire the beauty of Completing The Square.

Also,
Maths is not just about a whole bunch of formula only!

I met a lot of people who only memorized and just relied on formula,
after they graduated,
they looked for the formula to be successful in life,
only to find out that LIFE IS NOT JUST ABOUT FORMULA ONLY.

Oooops......
Let's get back to today's topic.

Before start learning about CTS (Completing The Square),
we need to mention about Perfect Square.

What is Perfect Sqaure?

Well,
let us look at some examples of Perfect Square Number:

4, 9, 16 are all Perfect Square Number,
because those numbers are from the multiplication of the same number.

Next,
let's have a look at examples of Perfect Square Quadratic:

The above 3 quadratics are all Perfect Square Quadratic,
since it all came from the multiplication of the same algebra.

Do everyone still remember about this formula that we learnt at secondary lower syllabus:

Looks complicated.
If you don't really understand the formula,
it will be a little hard to memorize it.

Well actually if you really understand it,
you will remember it automatically.

Let me explain it:
(If we do the expansion normally without using the formula above)

For the 1st example,
x+3 can be seen as x and +3,
then x multiply with x, +3 multiply with +3,
thus we get the red parts.

Then x multiply with +3, add itself again,
thus we get the green part which is +6x.

Since we were adding the same +3x,
so we can do it as x multiply with +3, then multiply with 2 to get the +6x.

That's why 2ab is at the middle of the formula.

You will automatically remember the formula if you really understand it:

For the 2nd example,
just look at the x and -4 separately,
then x square, -4x multiply with 2 get -8x, lastly the square of -4 is 16.

DONE.

No need memorize also able to expand it within one line.

Same like always,
expand the 10 questions below each in one line:

Answer:

All the 10 answers above are all Perfect Square Quadratic.

After learning about the Expansion of Perfect Square Quadratic,
is time to learn about Factorisation too.

Example:

Answer:

First,
we find a which is the easiest.

Then only we look for b.

Different than a,
we can't just use the 9 at the most behind to find b,
because square root of 9 can be 3 or -3.

So we use -6x which is at the middle.

And we are done with the Factorisation of Perfect Square Quadratic.

Of course,
above all are just explanation,
we don't have to write until so complicated when answering the question.

Basically we can just do it within one line:

Looking for a is very simple,
when x^2 don't have any number with it, just x^2 only,
then a is x.

After knowing that a is x,
then it is easy to find b!

Just use the +8 at the middle to divide by 2, and get +4,
then square root the 16 at behind, and get +4 or -4,
combining both,
we can conclude that b is +4.

Easy~

One more example:

Answer:

Lastly,
10 questions for exercise:

No answer will be provided.

Just use the expansion to expand your answer to check whether did you get back the same thing as the question.

That's all for today,
we will start learning about CTS in next chapter.

Thank you!

End.

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Related articles:
Before learning Factorisation...
Factorisation part.A
Factorisation part.B
Factorisation part.C


Important Links:
CONTACT ME
ABOUT ME
MY MATHS BLOG(Chinese ver.)
MY YOUTUBE


Feel free to e-mail me if you have any problems:
leonleongmaths@gmail.com

QUADRATIC EQUATIONS : How to do Factorisation part.C (SPM, IGCSE, A-LEVEL...)

(Since my time is limited, so for every chapter I will skip some and only focus on parts which most of the students found problem with it. Maybe after I retire, then only I will complete those skipped parts.)

From now on,
we will enter part.C,
which is the last part of Factorisation.

First let's have a look at a question example:

Don't worry,
the way of doing is still the same.

Answer:

Since the 2 at the front can only come from 1x2,
so is 2x & x.

Since the 2 at the front is positive,
so 2x & x must be positive too.

Since 2x need to multiply with +1 to get +2x,
so 2x and +1 cannot be in the same bracket.

Same,
in order for x to multiply with -1 to get -x,
x and -1 cannot be in the same bracket also.

Thus:

A simplified version of the answer:

let's look at one more example:

Answer:

As usual,
the simplified version of answer:

One more example:

Answer:

Or you can use the below method:

Let's practice 10 questions before we entering the final stage:
：

No answer will be provided,
just expand it back to check!

Alright,
let's move on to the final stage.

Example:

Answer:

Which combination of numbers to choose from?

What we can rely now,
is our experience of doing Maths for all these years,
and our Maths sense.

If we picked the wrong combination of numbers,
we will realize something is wrong at the middle,
the most important thing is don't give up,
and choose another combination of numbers to do.

This time,
I choose 2x & 3x,
and 3&5.

Remember to expand back,
this is the best way to confirm your answer.

I met with students who are too lazy and unwilling to expand back to check answer,
which is not encouraged at all.

The simplified version of answer:

In order to be pro with Factorisation,
just remember that:

Expand back to check the answer!
Expand back to check the answer!
Expand back to check the answer!

Check by yourself,
find out what's wrong by yourself,
improve by yourself.

Never give up,
keep trying until you get the correct answer!

I also kept learning from my own mistake,
to be better at Factorisation.

Talked too much,
let's look at the last example:

Answer:

Done.

The last 10 questions for practicing:

No answer will be provided,
just expand it back to check.

That's all for the Factorisation,
we will be entering Completing The Square at next chapter.

Frankly speaking,
I am not very satisfied with this few chapters of Factorisation.

I feel like there is still a big gap between teaching directly comparing with teaching through the blog,
hopefully I will be able to improve and do it better next time.

Feel free to comment if you have any question or any parts which you don't understand.

Of course,
even if you understand all these,
it doesn't mean that you can get full marks in your exam.

UNDERSTANDING is just UNDERSTANDING,
SOLVING QUESTION is different.

At least if you understood,
it will help you a lot in solving questions.

Maths should be learnt easily,
Practice should be done accordingly,
exam should be faced seriously.

Thank you!

End.

Please LIKE MY Facebook Page & FOLLOW MY TWITTER to be updated with the blog.


Related articles:
Before learning Factorisation...
Factorisation part.A
Factorisation part.B


Important Links:
CONTACT ME
ABOUT ME
MY MATHS BLOG(Chinese ver.)
MY YOUTUBE


Feel free to e-mail me if you have any problems:
leonleongmaths@gmail.com