Register to reply

Intersection points

by SeReNiTy
Tags: intersection, points
Share this thread:
SeReNiTy
#1
Aug7-05, 05:35 AM
P: 171
Hi guys, i'm just wondering is it possible to solve the following using algebra to obtain the points of intersection of the two curves f(x) = 6sqrt(x) and
g(x) = [(x+5)^2]/36

I got to the point where i reconized that the inverse of g(x) = 6sqrt(x) - 5 which looks alot like the function f(x), any hints or solutions to this problem?
Phys.Org News Partner Science news on Phys.org
Fungus deadly to AIDS patients found to grow on trees
Canola genome sequence reveals evolutionary 'love triangle'
Scientists uncover clues to role of magnetism in iron-based superconductors
HallsofIvy
#2
Aug7-05, 05:44 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,502
Yes, the inverse of g is f(x)- 5. I don't know that that helps a lot in finding points of intersection.

The only way I see of finding the points of intersection is to set f(x)= g(x), square both sides to get rid of the square root, and solve the resulting fourth degree equation
TD
#3
Aug7-05, 05:45 AM
HW Helper
P: 1,021
I don't know if you can get there your way (something with the inverse) but tehcnically, it's solvable since you get a 4th-degree polynomial. It won't be 'fun' though. Of course, watch out for introducing solution when squaring.

[tex]\frac{{\left( {x + 5} \right)^2 }}
{{36}} = 6\sqrt x \Leftrightarrow \left( {\frac{{\left( {x + 5} \right)^2 }}
{{36}}} \right)^2 = \left( {6\sqrt x } \right)^2 \Leftrightarrow \frac{{\left( {x + 5} \right)^4 }}
{{1296}} - 36x = 0[/tex]

If you'd want to know, mathematica gives me:

[tex]\begin{gathered}-5 + \frac{{\sqrt{\frac{{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}}}{3} +
72\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}}}}}{2} - \hfill \\
\frac{{\sqrt{\frac{-{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}} -
216\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}} +
\frac{279936}
{{\sqrt{\frac{{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}}}{3} +
72\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}}}}}}
{3}}}}{2}\end{gathered} [/tex]

and

[tex]\begin{gathered}-5 + \frac{{\sqrt{\frac{{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}}}{3} +
72\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}}}}}{2} + \hfill \\
\frac{{\sqrt{\frac{-{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}} -
216\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}} +
\frac{279936}
{{\sqrt{\frac{{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}}}{3} +
72\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}}}}}}
{3}}}}{2}\end{gathered} [/tex]

which are approx: [itex]x \to 0.013452\, \wedge \,x \to 29.150[/itex]

SeReNiTy
#4
Aug7-05, 05:48 AM
P: 171
Intersection points

Quote Quote by HallsofIvy
Yes, the inverse of g is f(x)- 5. I don't know that that helps a lot in finding points of intersection.

The only way I see of finding the points of intersection is to set f(x)= g(x), square both sides to get rid of the square root, and solve the resulting fourth degree equation
Oh, with the inverse thing, i just remembered doing a question a while ago that involved finding the points of intersection between two inverse functions, it was much easier to computer the intersection between one of the functions and y = x since inverse always intersect along that line.

How do i go about solving the 4th degree polynomial?
TD
#5
Aug7-05, 05:54 AM
HW Helper
P: 1,021
Quote Quote by SeReNiTy
How do i go about solving the 4th degree polynomial?
Just as there are formula's for the quadratic and cubic, there also exists one for the 4th-degree, named Ferrari.

http://mathworld.wolfram.com/QuarticEquation.html


Register to reply

Related Discussions
Finding points of intersection Precalculus Mathematics Homework 13
Contour map and Intersection points Calculus & Beyond Homework 1
Points on the curve of intersection Calculus & Beyond Homework 1
Intersection coordinates are my points right? Introductory Physics Homework 2
Points of intersection General Math 1