How can you simplify the quadratic formula using completing the square?

[agentredlum]

And if your function contains a radical what are the derivatives going to give you? MORE RADICALS!

When i say radical i mean an nth root or a power (NOT n) of an nth root.

Look i gave an example where i believe method mentioned by micromass fails but my method works.

You gave an example where both methods fail, so you really didn't help his case.

I can't factor x^5 - x + 1

But i can factor x^4 + x^2 + 1 so i challenge you to write this as a product of 2 expressions without using complex numbers

You got the theorem you mention 99% correct. Them 2 proved you cannot solve certain quintics using a FINITE number of operations, but you can approximate the roots of any polynomial to any degree of accuracy using infinite series, or other methods started by Vieta and continued by Cauchy, Weirstrauss, Tsirnhause, Klien and many others too numerous to list.

 

[agentredlum]

Oh shucks...i just looked at post 15. THAT IS A GREAT TRICK!

I hope i have not offended you, i guess i got a little defensive.

You didn't send a picture so i have to use my BRAIN to decode TeX because my browser does not decode it.

In post 15 you prove an irrationl number to an irrational power can be rational and indeed an integer in that example! (If sqrt(2)^sqrt(2) is irrational then you can raise it to an irrational power and turn it into 2. Either way your claim is confirmed)

Well done!

PLEASE POST MORE TRICKS!

 

[dodo]

I wonder why do you insist on your browser "not decoding TeX". The formulas are turned into images on the server side; there is no such thing as an "incapable browser" in this sense.

If you are seeing broken or repeated images, you may want to try clearing the browser's cache and then reloading the page. Hope this helps!

 

[agentredlum]

x^5 - x + 1 = 0 cannot be solved using a finite number of radicals but it can be solved using an INFINITE number of radicals. I remember reading about this in a number theory book. If someone can confirm or deny I would be greatfull.

 

[agentredlum]

I wonder why do you insist on your browser "not decoding TeX". The formulas are turned into images on the server side; there is no such thing as an "incapable browser" in this sense.

If you are seeing broken or repeated images, you may want to try clearing the browser's cache and then reloading the page. Hope this helps!

I am using Playstation 3, not a computer and it has limitations. For another example, i can't read pdf files.

 

[pwsnafu]

And if your function contains a radical what are the derivatives going to give you? MORE RADICALS!

When i say radical i mean an nth root or a power (NOT n) of an nth root.

You said polynomial previously. Are you talking about irrational coefficients?

Look i gave an example where i believe method mentioned by micromass fails but my method works.

You gave an example where both methods fail, so you really didn't help his case.

This threadjack started because of misapplication of square roots, specifically trying to apply (a^b)^c = a^bc to complex numbers, waaaay back in post #64. And it exploded from there.

I can't factor x^5 - x + 1

But i can factor x^4 + x^2 + 1 so i challenge you to write this as a product of 2 expressions without using complex numbers

Do you mean x⁴+x²+1 = (x²+x+1)(x²-x+1)?
You do this kind of thing in ring theory. We say "show x⁴+x²+1 is not irreducible on Z[X]". Standard homework question.

You got the theorem you mention 99% correct. Them 2 proved you cannot solve certain quintics using a FINITE number of operations, but you can approximate the roots of any polynomial to any degree of accuracy using infinite series, or other methods started by Vieta and continued by Cauchy, Weirstrauss, Tsirnhause, Klien and many others too numerous to list.

Yup. Technically still wrong. You can't express certain roots using finite number of addition, subtraction, multiplication, division and nth root. If you include more operations, you're fine.

Anyway, infinite series and approximation techniques is what analysis is about. So you are doing real analysis. Kinda my point.

 

[agentredlum]

You said polynomial previously. Are you talking about irrational coefficients?
This threadjack started because of misapplication of square roots, specifically trying to apply (a^b)^c = a^bc to complex numbers, waaaay back in post #64. And it exploded from there.
Do you mean x⁴+x²+1 = (x²+x+1)(x²-x+1)?
You do this kind of thing in ring theory. We say "show x⁴+x²+1 is not irreducible on Z[X]". Standard homework question.
Yup. Technically still wrong. You can't express certain roots using finite number of addition, subtraction, multiplication, division and nth root. If you include more operations, you're fine.

Anyway, infinite series and approximation techniques is what analysis is about. So you are doing real analysis. Kinda my point.

Yes! I am pleasantly surprised at how quickly you factored x^4 + x^2 + 1

This is a thread about math tricks (see title) and i know a trick to factoring x^4 + x^2 + 1

step 1 Add zero to x^4 + x^2 + 1 in the form (x - x) and re-arrange these 2 parts that equal zero in a clever way.

x^4 + x + x^2 - x + 1

step 2 Group the first 2 terms and the last 3 terms

(x^4 + x) + (x^2 - x + 1)

step 3 factor the first group

x(x^3 + 1) + (x^2 - x + 1)

x(x + 1)(x^2 - x + 1) + (x^2 - x + 1)

now you have a common factor of (x^2 - x + 1) I have used the sum of two cubes factoring technique,

(x^2 - x + 1)[x(x + 1) + 1]

step 4 simplify inside brackets

(x^2 - x + 1)(x^2 + x + 1)

There it is, in my opinion almost like magic, neat huh?

P.S. Only rational coefficients and i see your trap about the derivative, very clever sir. Whats the point of arguing further unless i am VERY mistaken. Let's be friends and post more tricks.

 

[agentredlum]

Anyone with some algebra skill can use (x - x) to derive factoring sum of two cubes formula and diference of two cubes formula.

I INVITE YOU ALL TO TRY!

P.S. I used to be able to factor certain trinomials of the 5th degree using carefully placed zero. Now I'm trying to remember which ones and how...

 

[pwsnafu]

Yes! I am pleasantly surprised at how quickly you factored x^4 + x^2 + 1

This is a thread about math tricks (see title) and i know a trick to factoring x^4 + x^2 + 1

This is how I did it: I know that there is no linear factor, so it's quadratic times quadratic. The coefficient of X⁴ is 1, and the constant term is 1. So one of the factors is X² + a X + 1. And just try different a. Naturally you try a=1 first. And you are done.

While we are on the topic of "no complex number tricks", when a student first learns complex numbers I give them this little gem:

Suppose x + y = 2 and xy = 3. Find 1/x + 1/y.

Nine times out ten, the student will try to use the quadratic formula. If the student doesn't know about complex numbers they'll stop and try to find a different method. The students that do know complex numbers will just keep going. :rofl:

 

[micromass]

micromass asked...

"Well, how would you solve $x^2=2+i$?? By saying $x=\sqrt{2+i}$?? How helpful..."

I would throw caution to the wind and let the chips fall where they may...THEN...

I would put this particular example into the quadratic formula and then sustitute back to confirm i got the right answer. Messere Gauss proved I can do it this way.

By all means, try it. You'll end up with $x=\sqrt{2+i}$. But that doesn't really help if you want a solution in the form $a+bi$

If you post a more complicated example which does not have a closed form solution i can still get close to the answer by using the 4 binary operations AND THE EXTRACTION OF ROOTS.

The ability to extract roots of real numbers is necessary, this means you need to extract roots of negative values.

The square root symbol has nothing to do with polynomials, really nothing. The reason why we can factor polynomials is because we introduced a number i such that $i^2=-1$. Wheter we write $\sqrt{-1}=i$ is irrelevant.

The Real numbers are 1 dimensional, Complex numbers NOT on the vertical axis have 2 distinct dimensions.

How fortunate that i is on the vertical axis and has length 1? So we can use it as a 'cheat' to simplify the algebra.

You are correct as far as the cyclotomic x^n = a + bi is concerned

However using sines and cosines to get the answer of course produces multi values since they are periodic!

No, I only get n values.

This is a neat trick, and i have solved x^n = a + bi in Linear Algebra so i know what you mean and agree with you on most of your points.

micromass also said...

The square root symbol has nothing to do with finding roots of polynomials. I can easily find the roots of $x^n=-1$:

To the best of my knowledge this method you describe works only if you don't have a MIDDLE TERM.

x^2 = (1 + 2i)x + i would not work with this method but easily works using quadratic formula.

Yes, using the quadratic formula is fine. But using the square root symbol is not necessary in the quadratic formula!

 

[agentredlum]

$${(}{{i}^{4}{)}}^{1/2}{=}{(}{\sqrt{i}{)}}^{4}$$

You agree to this[Your rule does not suggest this unless you incorporate some exception handling clause]?

What about [(-4)^(1/2)]^6 = [(-4)^6]^(1/2)]

The left is definitely-64 if you wish to avoid i, multiply the exponents.The right is not definitely -64 because you can simplify inside the brackets without fear of complex. As a matter of fact the right is +64 by convention but the left is -64 no matter how you evaluate it, so exchanging exponents like that is dangerous even for real negative numbers, let alone i and -i

 

[agentredlum]

This is how I did it: I know that there is no linear factor, so it's quadratic times quadratic. The coefficient of X⁴ is 1, and the constant term is 1. So one of the factors is X² + a X + 1. And just try different a. Naturally you try a=1 first. And you are done.

While we are on the topic of "no complex number tricks", when a student first learns complex numbers I give them this little gem:

Suppose x + y = 2 and xy = 3. Find 1/x + 1/y.

Nine times out ten, the student will try to use the quadratic formula. If the student doesn't know about complex numbers they'll stop and try to find a different method. The students that do know complex numbers will just keep going. :rofl:

I don't need quadratic formula or complex numbers for this, the answer is 2/3

I did it in my head in 5 billion nano-seconds

 

[agentredlum]

By all means, try it. You'll end up with $x=\sqrt{2+i}$. But that doesn't really help if you want a solution in the form $a+bi$
The square root symbol has nothing to do with polynomials, really nothing. The reason why we can factor polynomials is because we introduced a number i such that $i^2=-1$. Wheter we write $\sqrt{-1}=i$ is irrelevant.Yes, using the quadratic formula is fine. But using the square root symbol is not necessary in the quadratic formula!

Yes you are correct i suppose it's like solving x^2 = 4 and expecting a different result by using quadratic formula. That was silly of me, i should have tried it first. However that is not what i considered my important point, and I've been up all night and i can't remember what my point was/is anymore.

Please show me this quadratic formula you speak of that has no radicals.

 

[micromass]

Yes you are correct i suppose it's like solving x^2 = 4 and expecting a different result by using quadratic formula. That was silly of me, i should have tried it first. However that is not what i considered my important point, and I've been up all night and i can't remember what my point was/is anymore.

Please show me this quadratic formula you speak of that has no radicals.

Let $ax^2+bx+c=0$ be your equation. Let $D=b^2-4ac$, consider $Z$ such that $Z^2=D$, then the solutions are

$$x=\frac{-b\pm Z}{2a}$$

See, no radicals involved, but you still have the same formula This shows that you don't need the square root symbol anywhere...

 

[agentredlum]

Let $ax^2+bx+c=0$ be your equation. Let $D=b^2-4ac$, consider $Z$ such that $Z^2=D$, then the solutions are

$$x=\frac{-b\pm Z}{2a}$$

See, no radicals involved, but you still have the same formula This shows that you don't need the square root symbol anywhere...

Please post picture, I CAN'T READ IT!

it comes out as /frac-b/pm Z gibberish in my browser

 

[agentredlum]

Anyway, what is Z in your equation...sqrt(D)

nice try throwing smoke bomb but please, you have to do better than that.

agentredlum-passed out...

 

[micromass]

Anyway, what is Z in your equation...sqrt(D)

nice try throwing smoke bomb but please, you have to do better than that.

agentredlum-passed out...

Like I said, writing $\sqrt{D}=Z$ is not allowed since D is complex. But writing $Z^2=D$ is allowed here. I'm not doing anything new or fishy here I'm just arguing on when to use which symbols...

 

[agentredlum]

Like I said, writing $\sqrt{D}=Z$ is not allowed since D is complex. But writing $Z^2=D$ is allowed here. I'm not doing anything new or fishy here I'm just arguing on when to use which symbols...

Yes but your formula doesn't give any answers until you take the square root, a square roooooot you do not allow yourself to take.

 

[agentredlum]

How many hours you been without sleep?

 

[micromass]

Yes but your formula doesn't give any answers until you take the square root, a square roooooot you do not allow yourself to take, who's silly now?

I don't need to use the square root symbol anywhere. Where do you think I need it??

And I don't appreciate being called silly. If that's your attitude, then I don't think this discussion will go on for much longer.

How many hours you been without sleep?

I just woke up.

 

[agentredlum]

I don't need to use the square root symbol anywhere. Where do you think I need it??

And I don't appreciate being called silly. If that's your attitude, then I don't think this discussion will go on for much longer.
I just woke up.

I apologize, forget i said it.

pwsnafu is still trying to figure out how i got the answer.

 

[micromass]

I apologize, forget i said it.

pwsnafu is still trying to figure out how i got the answer.

I think pwsnafy know very well how you got the answer. It's the first thing you should try...

 

[agentredlum]

I edited the post removing the remark.

 

[agentredlum]

I think pwsnafy know very well how you got the answer. It's the first thing you should try...

And what is that?

 

[pwsnafu]

I apologize, forget i said it.

pwsnafu is still trying to figure out how i got the answer.

I think pwsnafy know very well how you got the answer. It's the first thing you should try...

And what is that?

Wait what? I didn't know I supposed to work something out. :tongue:

If we are talking about the 1/x+1/y problem, then yes, I know how to get the answer easily that's the whole point. Too often students don't analyze the question. If all you have is the quadratic formula, then everything is a quadratic equation. Especially if you're a high school student.

If we are not talking about that, what question?

 

[agentredlum]

Wait what? I didn't know I supposed to work something out. :tongue:

If we are talking about the 1/x+1/y problem, then yes, I know how to get the answer easily that's the whole point. Too often students don't analyze the question. If all you have is the quadratic formula, then everything is a quadratic equation. Especially if you're a high school student.

If we are not talking about that, what question?

I know what i was talking about, i was talking about that.

I don't know what micromass was talking about, i can't read minds. It appears to me that micromass wants to explain the radical in the quadratic formula is not necessary to numerically determine the roots using the quadratic formula.

 

[dodo]

(We interrupt this program for a short message...)

According to this PS3 manual,
http://manuals.playstation.net/document/en/ps3/current/browser/menub.html

In your "Network" menu, under "Tools" you can find an option to turn Javascript "on" or "off". If Javascript is "off", then that is the reason why you're seeing formulas as gibberish. Give it a try!

 

[agentredlum]

(We interrupt this program for a short message...)

According to this PS3 manual,
http://manuals.playstation.net/document/en/ps3/current/browser/menub.html

In your "Network" menu, under "Tools" you can find an option to turn Javascript "on" or "off". If Javascript is "off", then that is the reason why you're seeing formulas as gibberish. Give it a try!

I tried it just now, it didn't work. Thank you anyway.

If I delete cache will i lose my bookmarks? I have 1000 bookmarks many about math and science so I'm afraid of losing them. If you can assist i would be greatful.

 

[dimension10]

For trignometric equalitites, https://www.physicsforums.com/showthread.php?t=514065.

 

[Guffel]

I tried it just now, it didn't work. Thank you anyway.

If I delete cache will i lose my bookmarks? I have 1000 bookmarks many about math and science so I'm afraid of losing them. If you can assist i would be greatful.

No, you won't lose your bookmarks. Automatic logins on forums etc may disappear, but as long as you remember your user name and password, there should be no problem.

 

[Anamitra]

Well, how would you solve $x^2=2+i$?? By saying $x=\sqrt{2+i}$?? How helpful...

The square root symbol has nothing to do with finding roots of polynomials. I can easily find the roots of $x^n=-1$:

$$\cos(\pi k /n)+i\sin(\pi k /n),~k\in \{0,...,n-1\}$$

but by your method, you would only find $x=\sqrt[n]{x}$ (even if we would allow the square root on complex numbers). That's not really what we want, is it??

A Direct Application of the Square-Root Operator
[ By using the Binomial Theorem]
$${(}{2}{+}{i}{)}^{1/2}$$ ------------ [Expression 1]
$${=}{2}^{1/2}{(}{1}{+}{i/2}{)}^{1/2}$$
$${=}{2}^{1/2}{(}{1}{+}{\Sigma}{(}{-1}{)}^{r-1}\frac{{(}{2r-2}{)}{!}}{{2}^{3r-1}{r}{!}{(}{r-1}{)}{!}}{i}^{r}{)}$$ ----------- [Expression 2]
[In the above expression r=1,2,3,4... to infinity]
=[1+1/32-5/2048+21/65536-429/8388608 ……………]+i [1/4-1/128+7/8192-33/262144+...]
=1.4553 + 0.3435 i
The convergence of the two series may be established by considering[for each series] the absolute values of the terms and then by applying D’Alembert’s ratio test or some other suitable test of convergence. The general term given in the third step(relation(2)) of the calculation proves useful in this respect. [One may consider the ratio between alternate terms[their absolute values] in the summation in Expression 2 $${\mid}\frac{{u}_{r+2}}{{u}_{r}}{\mid}$$ as r tends to infinity]

Suppose we are to find:
$${{(}2+i{)}}^{1/5}$$
We find the first root by using the binomial theorem . Let the root be a+ib
$${y}^{5}{=}{2+i}$$
$${y}^{4}{=}\frac{2+i}{a+ib}$$
After rationalizing the RHS we take the fourth root by applying the binomial theorem again.
The process is repeated---- we do not get more than five roots by this method.

De Moivre’s method is much more convenient for these evaluations. But the application of the square root and other roots [their validity]can be seen from the above calculations.

 

[agentredlum]

No, you won't lose your bookmarks. Automatic logins on forums etc may disappear, but as long as you remember your user name and password, there should be no problem.

I tried it, still didn't work. Thank you very much for your concern.

 

[Char. Limit]

Here's something interesting. Consider the equation:

$$a^2 + b^2 + c^2 + d^2 = e$$

Now, let e be any natural number (here, referring to any positive integer OR zero). Then a, b, c, and d have solutions in the natural numbers. In other words, any natural number can be rewritten as the sum of four square numbers.

This result is known as Four-Square Theorem[/url].

 

[agentredlum]

Here's something interesting. Consider the equation:

$$a^2 + b^2 + c^2 + d^2 = e$$

Now, let e be any natural number (here, referring to any positive integer OR zero). Then a, b, c, and d have solutions in the natural numbers. In other words, any natural number can be rewritten as the sum of four square numbers.

This result is known as Four-Square Theorem[/url].

Thanx for the post, PLEASE POST MORE!

I find this interesting as well, Lagrange was a mathematical and science genius.

 

[agentredlum]

No one is posting tricks *sniff* let me ask some questions that I believe have fabulous Answers. I won't give the answers right away so you can think about it.

1) Using all digits 0~9 only once how many different numbers can you make?

Example: The number 1234567890 is one possibility, the number 1023456789 is another possibility. No digit is to be repeated anywhere in the number and all digits must be used.

2) How many of these numbers asked for above are divisible by 9?

3) How many zeroes are at the end of 1000!

Example: 7! = 5040 so there is one zero at the end of 7!

Good Luck