A good problem

Find no. of solutions of $\large(x^2-7x+11)^{(x-6)(x-5)}=x$

Please upload solution if u can

#Algebra

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Comments

Taking log both sides,

$(x-5)(x-6)\ln (x^2-7x+11)=\ln x$

Note that $x=5$ is a repeated root so the differential would be 0 at $x=5$ . Try to plot the graph. At $x=\infty,\ LHS=\infty$ . At $x=5,6,2$ , $LHS=0$ . At $x=0,\ LHS>0$ . I think graphs can be plotted now and it will intersect $y=\ln x$ at two points.

Pranjal Jain - 6 years, 5 months ago

@Chew-Seong Cheong @Aneesh Kundu @Pranjal Jain @megh choksi @Jon Haussmann @Trevor Arashiro @Trevor B. @Calvin Lin @Aman Sharma Please upload solution if u can

Mehul Chaturvedi - 6 years, 5 months ago

I'll post more tomorrow, but for now, I'd just say to create a sign chart of where the exponent and where the $x^2-7x+11$ are positive and when negative and where the LHS exceeds the RHS. The problem is that if the LHS intersects the RHS when negative, You have to watch for extraneous solutions because $(-2)^{\frac{7}{12}}$ is not an answer since its imaginary (the answer isn't 2^7/12, Im just using it as an example.

Trevor Arashiro - 6 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$