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October 1st, 2017, 11:13 AM  #1 
Member Joined: Sep 2014 From: Sweden Posts: 80 Thanks: 0  Simpler solve for this equation: sqrt(x^2x10)sqrt(x^211x)=10
Hi, I have solved this equation but it seems like there might be a simpler way to do it (or it is not a simpler way at all ), but however the equation that I have solved looks like this: $\displaystyle \displaystyle \sqrt{x^2x10}\sqrt{x^211x}=10\\\\\\\\ \displaystyle \sqrt{x^2x10}=10+\sqrt{x^211x}\\\\\\\\ \displaystyle x^2x10=100+20\sqrt{x^211x}+x^211x\\\\\\\\ \displaystyle 10x20\sqrt{x^211x}=110\\\\\\\\ \displaystyle 10(x2\sqrt{x^211x})=11\\\\\\\\ \displaystyle x11=2\sqrt{x^211x}\\\\\\\\ \displaystyle x^222x+121=4(x^211)\\\\\\\\ \displaystyle x^222x+121=4x^244x\\\\\\\\ \displaystyle 3x^2+22x+121=0\\\\\\\\ $ Use the abcformula $\displaystyle \displaystyle x=\frac{b\pm\sqrt{b^24ac}}{2a}\\\\\\\\$ $\displaystyle \displaystyle x=\frac{22\pm\sqrt{(22)^24\cdot 3\cdot 121}}{2\cdot 3}\\\\\\\\ \displaystyle x=\frac{22\pm\sqrt{484+1452}}{6}\\\\\\\\ \displaystyle x=\frac{22\pm\sqrt{1936}}{6}\\\\\\\\ \displaystyle x=\frac{22\pm44}{6}\\\\\\\\ $ The first solution (false) $\displaystyle \displaystyle x_1=\frac{22+44}{6}\\\\\\\\ \displaystyle x_1=\frac{22}{6}\\\\\\\\ \displaystyle x_1=\frac{11}{3} \ false\\\\\\\\ $ The second solution (true) $\displaystyle \displaystyle x_2=\frac{2244}{6}\\\\\\\\ \displaystyle x_2=\frac{66}{6}\\\\\\\\ \displaystyle x_2=\frac{66}{6}\\\\\\\\ \displaystyle x_2=11 \ true\\\\\\\\ $ Last edited by DecoratorFawn82; October 1st, 2017 at 11:16 AM. 
October 1st, 2017, 01:35 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,528 Thanks: 589 
There is no simpler way. Squaring steps introduce spurious roots. Fourth line has an error, but it was corrected for the fifth line.

October 1st, 2017, 06:03 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 19,065 Thanks: 1621 
Multiplying both sides by $\sqrt{x^2  x  10} + \sqrt{x^2  11x} + 10$, gathering like terms and dividing by 10 leads to $x  11 = 2\sqrt{x^2  11x}$, so $x^2  22x + 121 = 4x^2  44x$. Hence $3x^2  22x + 121 = 0$, i.e. $(x  11)(3x + 11) = 0$. As $x = 11/3$ doesn't satisfy the original equation, $x = 11$ is the unique solution. 

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equation, simpler, solve, sqrtx2x10sqrtx211x10 
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