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 April 12th, 2014, 09:19 PM #1 Senior Member   Joined: Apr 2013 Posts: 425 Thanks: 24 Another inequality To solve the inequality $\displaystyle x^4+log_23^{(x^2+4)}+5\leq0$. April 12th, 2014, 10:27 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,674 Thanks: 2654 Math Focus: Mainly analysis and algebra \begin{align*} x^4+log_23^{(x^2+4)}+5 &\leq 0 \\ x^4+\left( x^2 + 4 \right) \frac{\log{3}}{\log{2}} + 5 &\leq 0 \\ x^4 + \frac{\log{3}}{\log{2}} x^2 + \left(\frac{4\log{3}}{\log{2}} + 5\right) &\leq 0 \\ x^2 = \frac{1}{2}\left( -\frac{\log{3}}{\log{2}} \pm \sqrt{ \frac{\log^2{3}}{\log^2{2}} - 4\left(\frac{4\log{3}}{\log{2}} + 5\right) } \right) \end{align*} I reckon we have no real roots here, so I suppose that this is another one where we want to make the imaginary part of the equation disappear for complex $x$. Thus $x = a +b\imath$ \begin{align*} \operatorname{Im}{\left(x^4 + \frac{\log{3}}{\log{2}} x^2\right)} &= 0 \\ \operatorname{Im}{\left(a^4 + 4a^3b\imath - 6a^2b^2 -4ab^3\imath + b^4 + \frac{\log{3}}{\log{2}} \left(a^2 + 2ab\imath - b^2 \right) \right)} &= 0 \\ 4a^3b -4ab^3 + \frac{\log{3}}{\log{2}} 2ab &= 0 \\ \end{align*} Which gives $b = 0$ (which we have already dismissed) or $$b^2 = a^2 + \frac{\log{3}}{2\log{2}}$$ Which we substitute into \begin{align*} \operatorname{Re}{\left(a^4 + 4a^3b\imath - 6a^2b^2 -4ab^3\imath + b^4 + \frac{\log{3}}{\log{2}} \left(a^2 + 2ab\imath - b^2 \right) \right)} &\leq -\left(\frac{4\log{3}}{\log{2}} + 5\right) \\ a^4 - 6a^2b^2 + b^4 + \frac{\log{3}}{\log{2}} \left(a^2 - b^2 \right) &\leq -\left(\frac{4\log{3}}{\log{2}} + 5\right) \\ a^4 - 6a^2\left(a^2 + \frac{\log{3}}{2\log{2}}\right) + \left(a^2 + \frac{\log{3}}{2\log{2}}\right)^2 + \frac{\log{3}}{\log{2}} \left(a^2 - \left(a^2 + \frac{\log{3}}{2\log{2}}\right) \right) &\leq -\left(\frac{4\log{3}}{\log{2}} + 5\right) \\ -4a^4 -\frac{2\log{3}}{\log{2}}a^2 - \frac{\log^2{3}}{4\log^2{2}} &\leq -\left(\frac{4\log{3}}{\log{2}} + 5\right) \\ 4a^4 + \frac{2\log{3}}{\log{2}}a^2 + \frac{\log^2{3}}{4\log^2{2}} -\frac{4\log{3}}{\log{2}} - 5 &\ge 0 \\ \end{align*} This we can solve for a^2, but it's half-past one in the morning, I'm going to bed instead. I'd be very surprised if that algebra is without errors. Thanks from Olinguito Last edited by v8archie; April 12th, 2014 at 10:39 PM. April 13th, 2014, 05:55 AM   #3
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 Originally Posted by v8archie This we can solve for a^2, but it's half-past one in the morning, I'm going to bed instead. I'd be very surprised if that algebra is without errors.
I do not understand!For what values of $\displaystyle x$ is satisfied the inequality?

Last edited by Dacu; April 13th, 2014 at 06:28 AM. April 13th, 2014, 06:40 PM #4 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,674 Thanks: 2654 Math Focus: Mainly analysis and algebra I didn't go as far as getting the values of $x$ because the algebra is getting far too horrible. Suffice to say that $x$ will be complex, satisfying $x = a + b\imath$ where $a$ and $b$ satisfy equations given in my post above. If my efforts are correct, the question is phrased badly and should say something like "Find values of $z \in \mathbb{C}$ such that $z^4+log_23^{(z^2+4)}+5$ is real and less than or equal to zero." April 13th, 2014, 08:47 PM   #5
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Quote:
 Originally Posted by v8archie I didn't go as far as getting the values of $x$ because the algebra is getting far too horrible. Suffice to say that $x$ will be complex, satisfying $x = a + b\imath$ where $a$ and $b$ satisfy equations given in my post above. If my efforts are correct, the question is phrased badly and should say something like "Find values of $z \in \mathbb{C}$ such that $z^4+log_23^{(z^2+4)}+5$ is real and less than or equal to zero."
Any equation of any degree $\displaystyle a_nx^n+a_{n-1}x^{n-1}+\dots +a_1x+a_0=0$ can have real solutions and/or complex and so and any inequality of this type,because any inequality can turn into an equation. Tags inequality Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post eddybob123 Math Events 1 July 22nd, 2013 09:51 PM lambda Linear Algebra 3 September 8th, 2012 03:28 AM jatt-rockz Algebra 2 November 5th, 2011 08:21 PM JC Algebra 6 October 29th, 2011 04:32 PM salamatr25 Algebra 0 July 26th, 2010 04:00 AM

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