Core 1 - Questions involving "Square root"

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WOW! I am on FP2/FP3 and I had no idea, one thing though

\displaystyle x^2 = 4 \implies \sqrt {x^2} = \sqrt 4

and

\displaystyle \sqrt 4 = +2

as the

Unparseable latex formula:

\displaystyle \sqrt

function gives the positive value.

so how come

\displaystyle x = \pm 2

?

And if

\displaystyle x^2 = 7

then would

\displaystyle x = \sqrt7

\displaystyle \pm \sqrt 7

x^2 = 7 \implies x= \pm \sqrt7

x^2 = 4 \implies x = \pm \sqrt4 = \pm 2

\sqrt4 = 2 \not= -2

Do i make any sense?

Reply 41

member910132

Original post by hassi94

Not really sure what you're asking in your top line.

x^2 = 4 \implies x = \pm\sqrt{4}

What you wrote is correct but that is only one answer as you only took the positive square root of both sides.

and for the second bit,

\pm\sqrt{7}

This is strange tbh, we are saying that the rt 4 in x^2 = 4 is different to a random rt 4 we see elsewhere because the first one gives \pm 2 and the second gives just 2. I can accept it for the sake of exams but I have always loved maths for being logical and having answers for everything, which doesn't seem to be the case here.

Reply 42

Intriguing Alias

Original post by member910132

No you're mistaken. My point was that your post was only partly correct. x^2 = 4 does imply x = 2 if x had to be positive. But in reality it implies x = 2 OR x = -2

It's not just for the sake of exams; unless you write plus/minus then the square root sign JUST means to positive square root. It is perfectly logical and I'm not sure why you think otherwise. You used to think that writing

\sqrt{4}

was the same as

\pm\sqrt{4}

but now you know it means

+\sqrt{4}

. That's not really a big deal; in fact it's more logical as you would never assume something to be plus/minus unless it says so. Having no sign prefix has always meant positive in other situations (i.e. we write x^2 not +x^2 because that's implied by convention).

Reply 43

Mr M

Study Forum Helper

x^2=4

x^2-4=0

(x-2)(x+2)=0

x=\pm 2

Reply 44

raheem94

Original post by member910132

To be honest, nuodai is the best person to address this issue.

Though i will try to explain it a bit.

y = \sqrt4 = 2 \not= \pm 2

Lets see the graph of the function

y= \sqrt{x}

The graph always is in the positive regions hence we don't consider the negative solution.

For the case of

x^2 = 4 \implies x= \pm \sqrt4 = \pm 2

See the graph of

y=x^2

There are 2 values of

x

for every value of

y

2^2=4 \ \ and \ \ (-2)^2 = 4

Now lets look back at

\sqrt4 = 2 \ \ and \ \ \sqrt4 = -2

If we square both sides of these expression then both will be equal, but in this case we will be adding extra solutions.

e.g.

x=2

If we square both sides, we get,

x^2 = 4 \implies x= \pm 2

So squaring both sides adds extra solutions.

Do i make any sense?

Reply 45

member910132

Original post by Mr M

x^2=4

x^2-4=0

(x-2)(x+2)=0

x=\pm 2

lol, this is a great way of explaining it, if only I could see things like you :tongue:

Reply 46

member910132

Original post by raheem94

Do i make any sense?

Why Nudoi ?

You make perfect sense and I am glad I asked this, I understand this perfectly now, thanks to your time and effort in that post.

So in terms of general algebra, should we try and avoid squaring both sides as that will give rise to extra solutions ?

Reply 47

raheem94

Original post by member910132

I think Nuodai is the best when it comes to explaining in detail. In my experience, i haven't seen anyone better in explaining the stuff than Nuodai.

You should always try to avoid squaring, unless necessary, because it can add extra solutions.

To clarify the last post a bit more,

x= 2

We know that

2^2 =2

, while

(-2)^2 = 4

, so both are equal.

So if we square,

x= 2

, then it will also be giving the result for

x= -2

because squaring both

x= 2 \ \ and \ \ x = -2

gives the same equation.

(edited 11 years ago)

Reply 48

Intriguing Alias

Original post by member910132

Sometimes squaring both sides needs to be done (in cases where we have square roots of functions etc). But yes you must always be aware of adding extra solutions.

Jesus wept.

Original post by raheem94

I will get mad if i try to scroll back through your posts :mad:

I really didn't knew that people have confusion on this, when the other guy said he didn't knew it, i thought he is trying to troll.

So your teacher in year 9 wasn't clear about this concept as well, right?

I'm Studying a Mathematics degree and I didn't know of this either.

Reply 51

Mr Tough

Original post by Mr M

Oh dear.

Original post by raheem94

Did your teacher really told you this?

I didn't expected this from you.

I'm not saying it's correct mathematically, but after all those past papers, I can say for certain, that is the rule for Edexcel papers at least.

Original post by raheem94

This guy got 100 in both C1 and C2.

Just did M1 today, hoping for the same result :smile:

Original post by Mr M

I wouldn't worry about it. I saw this guy admit he had no idea about it a couple of weeks ago on twitter. He even tried to argue it was wrong.

http://cambridge.academia.edu/JamesGrime

No, whoever this guy is, he isn't me !