1)Solve this equation: x+√(x^2+4/x^2 +4)=4
2)a, b and c are the sides of a triangle and p , q are positive numbers and p+q = 1 . Prove that pa^2+qb^2≥pqc^2.
3)a and b are real positive numbers. a^n=a+1 and b^2n=b+3
n ∈N and n >1. Compare a with b.
1)Solve this equation: x+√(x^2+4/x^2 +4)=4
2)a, b and c are the sides of a triangle and p , q are positive numbers and p+q = 1 . Prove that pa^2+qb^2≥pqc^2.
3)a and b are real positive numbers. a^n=a+1 and b^2n=b+3
n ∈N and n >1. Compare a with b.
1. $\displaystyle x+\sqrt{x^2+\frac{4}{x^2}+4}=4$
$\displaystyle \sqrt{x^2+\frac{4}{x^2}+4}=4-x$
$\displaystyle x^2+\frac{4}{x^2}+4=(4-x)^2$
$\displaystyle x^2+\frac{4}{x^2}+4=16-8x+x^2$
$\displaystyle \frac{4}{x^2}+4=16$
$\displaystyle 4+4x^2=16x^2$
$\displaystyle 12x^2-4=0$
$\displaystyle 3x^2-1=0$
$\displaystyle x=\pm\sqrt{\frac{1}{3}}$
Assume
$\displaystyle pa^2+qb^2 < pqc^2$
Let c be the hypotenuse, since it is the longest side of a triangle, and therefore the greatest number.
Then by Pythagoreans theorem, $\displaystyle a^2+b^2=c^2$
We can substitute $\displaystyle a^2+b^2=c^2$ into our equation:
$\displaystyle pa^2+qb^2 < pq(a^2+b^2)$
And simplify:
$\displaystyle pa^2+qb^2 < pqa^2+pqb^2$
Now we know that p and q are between zero and one, thus pq < p and pq < q
Which means that:
$\displaystyle pqa^2<pa^2$
$\displaystyle pqb^2<qb^2$
So we can say that there are values $\displaystyle v_1 \mbox{ and } v_2$ which are greater than zero, where:
$\displaystyle pqa^2+v_1=pa^2$
$\displaystyle pqb^2+v_2=qb^2$
Now substituting these values into our equation:
$\displaystyle pqa^2+v_1+pqb^2+v_2 = pqa^2+pqb^2$
And subtracting $\displaystyle pqa^2 + pqb^2$ from both sides we get
$\displaystyle v_1+v_2 = 0$
Since $\displaystyle v_1 \mbox{ and } v_2$ are greater than zero, their sum is greater than zero:
$\displaystyle v_1+v_2 > 0$
but this contradicts that they equal zero.
$\displaystyle v_1+v_2 = 0$
So our assumption that $\displaystyle pqc^2$ is greater than $\displaystyle pa^2 + qb^2$ must be incorrect.
Therefore $\displaystyle c^2$ is not greater than $\displaystyle pa^2 + qb^2.$
So $\displaystyle c^2$ must be either less than or equal to $\displaystyle pa^2 + qb^2$
I graphed the first one, and found that x=1 is also a solution, I don't know how to find it mathematically, though, except with Newton's method.
edit:
Okay, I got it.
$\displaystyle x+\sqrt{x^2+\frac{4}{x^2}+4}=4$
$\displaystyle \sqrt{x^2+\frac{4}{x^2}+4}=4-x$
$\displaystyle x^2+\frac{4}{x^2}+4=16-8x+x^2$
$\displaystyle 8x + \frac{4}{x^2}+-12=0$
$\displaystyle 2x + \frac 1{x^2}-3=0$
$\displaystyle 2x^3 -3x^2 +1=0$
Now there is a formula, I think it was Gauss who found it, but it's been a few semesters so I can't remember it exactly. It says something along the lines of:
If you have a polynomial, and p is one of the factors of the coefficient of the first term (so I guess that would be $\displaystyle p\in \{1, 2, -1, -2\}$), and q is a factor of the coefficient of the second term (so $\displaystyle q\in \{1, 3, -1, -3\}$) then if there is a factor, it will take the form of $\displaystyle (x+\frac pq) \mbox{ or } (x+ \frac qp)$
In this case, lets use q/p, where q=1 and p=2
then I did long division (I forgot how to do short division >.<)
$\displaystyle \begin{array}{rr} &\begin{array}{r} 2x^2-4x+2 \end{array} \\ \begin{array}{r} \\(x+\frac 12) \end{array} & \begin{array}{|r} \hline\\ 2x^3-3x^2+0x+1 \end{array} \end{array}$
So $\displaystyle (x+1/2)$ is a factor, giving us
$\displaystyle (x+\frac 12)(2x^2 -4x +2)=0$
$\displaystyle \frac 12(x+\frac 12)(x^2 -2x +1)=0$
$\displaystyle (x+\frac 12)(x -1)^2=0$
So x= -.5 or x=1
Hello, blertta!
1) Solve: .$\displaystyle x+ \sqrt{x^2+\frac{4}{x^2} +4}\:=\:4 $
We have: .$\displaystyle \sqrt{x^2 + \frac{4}{x^2} + 4} \;=\;4 - x$
Square both sides: .$\displaystyle x^2 + \frac{4}{x^2} + 4 \;=\;16 - 8x + x^2 \quad\Rightarrow\quad 8x - 12 + \frac{4}{x^2} \;=\;0
$
Multiply by $\displaystyle \frac{x^2}{4}\!:\;\;2x^3 - 3x^2 + 1 \;=\;0$
We find that $\displaystyle x = 1$ is a root.
The cubic factors: .$\displaystyle (x-1)^2(2x+1) \;=\;0$
Answers: . $\displaystyle \boxed{x \;=\;1,\:-\frac{1}{2}}$
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Did anyone notice that the radical contains a perfect square?
We have: .$\displaystyle x + \sqrt{x^2 + 4 + \frac{4}{x^2}} \;=\;4\quad\Rightarrow\quad x + \sqrt{\left(x+\frac{2}{x}\right)^2} \;=\;4$
Then: .$\displaystyle x \pm \left(x + \frac{2}{x}\right) \;=\;4$ . . . which yields the two roots above.
for division of polynomials the short version would be synthetic division. i don't know how to do that either. i'm sure it's easy, but never had any interest in learning it, i'm good and fast enough at long division of polynomials.
nice use of LaTeX here!$\displaystyle \begin{array}{rr} &\begin{array}{r} 2x^2-4x+2 \end{array} \\ \begin{array}{r} \\(x+\frac 12) \end{array} & \begin{array}{|r} \hline\\ 2x^3-3x^2+0x+1 \end{array} \end{array}$
(i wonder if i can improve on the code, maybe not. but there must be a way of not using so many arrays)
Hello, angel.white!
Well, okay, I do!I noticed that the terms were squares, but I didn't see they were a perfect square,
I don't have enough exposure to unusual formats. . . . . Who does?
To tell the truth, I learned about these interesting forms during Calculus
. . while working with Arc Length problems.
The arc length formula is: .$\displaystyle L \;=\;\int^b_a\sqrt{1 + \left(\frac{dy}{dx}\right)^2}\,dx$
We must take the derivative of the function, square it, add one, and take the square root.
. . Then we must integrate this ugly function.
If the problem's author is humane, the expression simplifies very nicely.
However, by necessity, the original function is usually strange-looking.
Here's an example: .$\displaystyle y \:=\:\frac{1}{6}x^3 + \frac{1}{2x} \;=\;\frac{1}{6}x^3 + \frac{1}{2}x^{-1}$
(Seriously, who would want to graph that curve anyway
. . and find the length of a portion of it?)
We have: .$\displaystyle \frac{dy}{dx} \:=\:\frac{1}{2}x^2 - \frac{1}{2}x^{-2} \:=\;\frac{1}{2}\left(x^2-\frac{1}{x^2}\right) $
Then: .$\displaystyle \left(\frac{dy}{dx}\right)^2 \;=\;\frac{1}{4}\left(x^2-\frac{1}{x^2}\right)^2 \;=\;\frac{1}{4}\left({\color{red}x^4 - 2 + \frac{1}{x^4}}\right)$ .[1]
And: .$\displaystyle 1 + \left(\frac{dy}{dx}\right)^2 \;=\;1 + \frac{1}{4}\left(x^4 - 2 + \frac{1}{x^4}\right)$
. . $\displaystyle = \;\frac{1}{4}\left(x^4 - 2 + \frac{1}{x^4} + 4\right) \;= \;\frac{1}{4}\left({\color{red}x^4 + 2 + \frac{1}{x^4}}\right)$
. . which looks like [1] except for a sign-change.
Hence, we are expected to recognize it as: .$\displaystyle \frac{1}{4}\left(x^2 + \frac{1}{x^2}\right)^2$
Finally: .$\displaystyle \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \;=\;\sqrt{\frac{1}{4}\left(x^2 + \frac{1}{x^2}\right)^2} \;=\;\frac{1}{2}\left(x^2 + \frac{1}{x^2}\right) $
And now we can integrate: .$\displaystyle L \;=\;\frac{1}{2}\int^b_a\left(x^2 + x^{-2}\right)\,dx$
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If you haven't had Calculus yet, it's still an interesting bit of algebra.
It's a great reputation-maker.
Glance at $\displaystyle x^8 + 6 + \frac{9}{x^8}$ and say: .Obviously, it's $\displaystyle \left(x^4 + \frac{3}{x^4}\right)^2$
. . and ride off into the sunset.
Blertta, I don't know if you saw this, but I got problem #2 in post 5: http://www.mathhelpforum.com/math-he...513-post5.html
Thank you, it's my first use of arrays ^_^