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 Algebra Pre-Algebra and Basic Algebra Math Forum

June 29th, 2018, 02:23 PM   #11
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Quote:
 Originally Posted by MathsLearner123 This is the attempt I have done. The final equation is $\displaystyle x(\lambda^3+4\lambda-40)=0;$ I am not sure why I got a different equation last time. Sorry if the attachments are not clear. Next time, I will try to use math editor. Please advise whether I am correct.
get rid of the factor of $x$ in front and you'll have it correct.

There is one real solution.

Last edited by skipjack; June 29th, 2018 at 09:32 PM. June 29th, 2018, 09:51 PM   #12
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Quote:
 Originally Posted by MathsLearner123 The final equation is $\displaystyle x(\lambda^3+4\lambda-40)=0$ Please advise whether I am correct.
That is the correct equation, but $y$ and $z$ can be eliminated without dividing by anything that might be zero.

Note that $\displaystyle \lambda^3 + 4\lambda - 40$ is the determinant of the matrix of coefficients of $x$, $y$ and $z$ in the original three equations.

Can you prove that $\displaystyle \lambda^3 + 4\lambda - 40 = 0$ has only one real solution? June 30th, 2018, 08:16 PM   #13
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Quote:
 Originally Posted by skipjack Can you prove that $\displaystyle \lambda^3 + 4\lambda - 40 = 0$ has only one real solution?
I tried from this https://en.wikipedia.org/wiki/Cubic_function
and calculated the discriminant which is less than 0 and hence it has one real root.
Discriminant is
$\displaystyle ax^3+bx^2+cx+d=0; \text{ discriminant } = 18abcd-4b^3d+b^2c^2-4ac^3-27a^2d^2; a=1, b=0, c=4,d=-40 =0 - 0 + 0 - 256 - 43200 =-43456$
Attached Images discriminant.jpg (20.2 KB, 2 views)

Last edited by skipjack; June 30th, 2018 at 11:01 PM. June 30th, 2018, 10:12 PM #14 Senior Member   Joined: Sep 2015 From: USA Posts: 2,553 Thanks: 1403 Perhaps a simpler way of showing it has only one real root is to realize that $\lambda^3+4\lambda -40$ is strictly increasing. Tags infinite, problem, solutions 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 blimper Linear Algebra 4 August 30th, 2016 06:53 PM king.oslo Linear Algebra 4 April 6th, 2014 04:47 PM 84grandmarquis Linear Algebra 2 September 29th, 2013 01:59 PM EduXx Computer Science 2 October 7th, 2012 03:10 PM 84grandmarquis Algebra 0 December 31st, 1969 04:00 PM

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