 My Math Forum Simpler solve for this equation: sqrt(x^2-x-10)-sqrt(x^2-11x)=10
 User Name Remember Me? Password

 Algebra Pre-Algebra and Basic Algebra Math Forum

 October 1st, 2017, 11:13 AM #1 Member   Joined: Sep 2014 From: Sweden Posts: 94 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: 6,704 Thanks: 670 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: 20,373 Thanks: 2010 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 Tags equation, simpler, solve, sqrtx2x10sqrtx211x10 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 Lalaluye Calculus 2 July 3rd, 2017 07:52 PM Married Math Calculus 7 September 19th, 2014 07:56 AM Singularity Calculus 4 October 24th, 2012 11:44 AM MathJustForFun Algebra 3 May 13th, 2010 01:03 AM nikkor180 Real Analysis 3 June 25th, 2009 03:18 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top      