My Math Forum Simpler solve for this equation: sqrt(x^2-x-10)-sqrt(x^2-11x)=10

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 October 1st, 2017, 11:13 AM #1 Member   Joined: Sep 2014 From: Sweden Posts: 75 Thanks: 0 Simpler solve for this equation: sqrt(x^2-x-10)-sqrt(x^2-11x)=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: $\displaystyle \displaystyle \sqrt{x^2-x-10}-\sqrt{x^2-11x}=10\\\\\\\\ \displaystyle \sqrt{x^2-x-10}=10+\sqrt{x^2-11x}\\\\\\\\ \displaystyle x^2-x-10=100+20\sqrt{x^2-11x}+x^2-11x\\\\\\\\ \displaystyle 10x-20\sqrt{x^2-11x}=110\\\\\\\\ \displaystyle 10(x-2\sqrt{x^2-11x})=11\\\\\\\\ \displaystyle x-11=2\sqrt{x^2-11x}\\\\\\\\ \displaystyle x^2-22x+121=4(x^2-11)\\\\\\\\ \displaystyle x^2-22x+121=4x^2-44x\\\\\\\\ \displaystyle -3x^2+22x+121=0\\\\\\\\$ Use the abc-formula $\displaystyle \displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\\\\\$ $\displaystyle \displaystyle x=\frac{-22\pm\sqrt{(-22)^2-4\cdot -3\cdot 121}}{2\cdot -3}\\\\\\\\ \displaystyle x=\frac{-22\pm\sqrt{484+1452}}{-6}\\\\\\\\ \displaystyle x=\frac{-22\pm\sqrt{1936}}{-6}\\\\\\\\ \displaystyle x=\frac{-22\pm44}{-6}\\\\\\\\$ 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) $\displaystyle \displaystyle x_2=\frac{-22-44}{-6}\\\\\\\\ \displaystyle x_2=\frac{-66}{-6}\\\\\\\\ \displaystyle x_2=\frac{66}{6}\\\\\\\\ \displaystyle x_2=11 \ true\\\\\\\\$ Last edited by DecoratorFawn82; October 1st, 2017 at 11:16 AM.
 October 1st, 2017, 01:35 PM #2 Global Moderator   Joined: May 2007 Posts: 6,454 Thanks: 567 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: 18,704 Thanks: 1529 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. Thanks from DecoratorFawn82

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