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Blog Post 51
EpicSkills32 1
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by briantix
![$\ [\text{Blog Post 51}] $](http://latex.artofproblemsolving.com/b/e/4/be4a401c8e76e3bf07a9e2840c316299239bf814.png)
So a few blog posts ago, I posted a problem.
Find all solutions to the following:
![\[ x^4-4x=1 \]](http://latex.artofproblemsolving.com/c/a/2/ca2bbb9c2eca7c33a344b745a7c678b658a9304a.png)
Oh dear. We have something to the fourth power. What's even worse, there isn't anything cubed or squared. hm....
At first, we see the 1 and the even power, and we do something like:
![\[ x^4-1=4x \]](http://latex.artofproblemsolving.com/e/2/a/e2aad0599a367324c525c1fa5132cb1b1469d153.png)
![\[ (x^2+1)(x^2-1)=4x \]](http://latex.artofproblemsolving.com/c/7/7/c77f81c851590051871d8bfb0d148d96925a8371.png)
![\[ (x^2+1)(x+1)(x-1)=4x \]](http://latex.artofproblemsolving.com/8/5/a/85a46eaf61e5830456a4464136595ea0b98ce73e.png)
And then we get stuck.
Stumped?
Instead, let's try to force a nicer polynomial.
We add
to both sides.![\[ x^4+2x^2-4x+1=2x^2+2 \]](http://latex.artofproblemsolving.com/6/d/e/6dea10506b9ed9576aa47a85136453e4cbce0766.png)
![\[ x^4+2x^2+1=2x^2+4x+2 \]](http://latex.artofproblemsolving.com/a/3/a/a3a0493e9e9f7177e965f67d65dfde9e1cb25429.png)
Aha! Something that can be factored!
![\[ (x^2+1)^2=2(x^2+2x+1) \]](http://latex.artofproblemsolving.com/3/a/1/3a18a3961583ae5a61ad901d4f18b1378914d046.png)
![\[ (x^2+1)^2=2(x+1)^2 \]](http://latex.artofproblemsolving.com/d/6/9/d693051ebf6c2e36ec602e57ce133841354343f8.png)
Oh but wait a second. The term on the left can be changed so that there is no coefficient in front.
![\[ 2(x+1)^2 = 2x^2+4x+2 = (\sqrt{2}x+\sqrt{2})^2 \]](http://latex.artofproblemsolving.com/5/8/e/58ee32e19aa282c2e8a569f74096b6438bbbcefb.png)
Now we can express our equation with two squared terms:
![\[ (x^2+1)^2=(\sqrt{2}x+\sqrt{2})^2 \]](http://latex.artofproblemsolving.com/2/c/1/2c1d68ffa99f1656849ac59027121961994448ca.png)
Since both sides are squared, we move both terms to one side to get a difference of squares:
![\[ (x^2+1)^2-(\sqrt{2}x+\sqrt{2})^2 \]](http://latex.artofproblemsolving.com/1/4/9/1493133ad1f5cfd15c227d933805941d2abf20e7.png)
Expanding:
\[ \left((x^2+1)+(\sqrt{2}x+\sqrt{2})\right)\left((x^2+1)-(\sqrt{2}x+\sqrt2})\right)= 0 \]![\[ (x^2+\sqrt{2}x+\sqrt{2}+1)(x^2-\sqrt{2}-\sqrt{2}+1) = 0 \]](http://latex.artofproblemsolving.com/1/a/d/1ad7dd68648a5411278b724d0e36bd81b21b1433.png)
We have two factors that we need to set equal to zero. Since they are just quadratics, we can do so with the quadratic formula:
![\[ (x^2+\sqrt{2}x+\sqrt{2}+1)=0 \]](http://latex.artofproblemsolving.com/3/0/3/303e5d0c549ffd4bfe906869754c7180e549225f.png)
![\[ a= 1 \]](http://latex.artofproblemsolving.com/8/b/3/8b3d754b5ffc3f063f594cd6cbffe3ae6c87683a.png)
![\[b=\sqrt{2}\]](http://latex.artofproblemsolving.com/f/d/1/fd133dcda3ecf02d2dc614ca259d072fb5eb5e86.png)
![\[c=\sqrt{2}+1\]](http://latex.artofproblemsolving.com/b/8/2/b8251f8e686be0c73a8bc0f441a8f03c0985d463.png)
![\[ \therefore x= \dfrac{-\sqrt{2}\pm\sqrt{2-4(\sqrt{2}+1)}}{2} \]](http://latex.artofproblemsolving.com/e/6/3/e63c50e4d4c4c48b4acd773179a78a08fea3b90b.png)
![\[ x= \dfrac{-\sqrt{2}\pm\sqrt{-2-4\sqrt{2}}}{2} \]](http://latex.artofproblemsolving.com/b/2/2/b226779a632ad7b8f2082d823e15e7a3fd41a683.png)
We have these ugly negatives under a terrible radical, so we change that:
![\[ x= \dfrac{-\sqrt{2}\pm i\sqrt{2+4\sqrt{2}}}{2} \]](http://latex.artofproblemsolving.com/a/1/1/a11f425d3723e284b111a5eba3c099eae0adfb5e.png)
Again for the other factor:
![\[ (x^2-\sqrt{2}-\sqrt{2}+1)=0 \]](http://latex.artofproblemsolving.com/f/5/9/f5919ae69c5f60e7841c4abe9cee9d06994dace1.png)
![\[ a=1 \]](http://latex.artofproblemsolving.com/1/c/0/1c09a1c3159700dc4ab005b27d82235d70001225.png)
![\[b=-\sqrt{2}\]](http://latex.artofproblemsolving.com/5/d/5/5d5d82b3524fd05cf933364fd81f82b1f5543194.png)
![\[c=-\sqrt{2}+1 \]](http://latex.artofproblemsolving.com/c/7/b/c7b22b216fe982e05bea66d602c275db567fa3b5.png)
![\[ \therefore x= \dfrac{\sqrt{2}\pm\sqrt{2-4(-\sqrt{2}+1)}}{2} \]](http://latex.artofproblemsolving.com/0/f/3/0f30f5017a0962ebdb8812672cfd00a050721460.png)
![\[ x= \dfrac{-\sqrt{2}\pm\sqrt{-2+4\sqrt{2}}}{2} \]](http://latex.artofproblemsolving.com/9/5/3/9538589223de7e5e9a314031213bc19ea7f28c21.png)
In the end, our solutions for
are the following values:![\[ x= \left(\dfrac{-\sqrt{2}\pm i\sqrt{2+4\sqrt{2}}}{2}\right),\left(\dfrac{-\sqrt{2}\pm\sqrt{-2+4\sqrt{2}}}{2}\right) \]](http://latex.artofproblemsolving.com/1/6/1/16117f5742e15577a09ddf1e2229ef3fc540e4ec.png)
Yay. :)
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