# find intersections without calculator

• Mar 14th 2009, 07:38 AM
gammaman
find intersections without calculator
How would I find complex intersections without the intersection function on the calculator.

Say I have y=4xe^(-x^2) and y = |x|. How would I find this intersection.

Note. I really only need the x values because I am trying to then find the area between these two curves.
• Mar 14th 2009, 08:14 AM
skeeter
Quote:

Originally Posted by gammaman
How would I find complex intersections without the intersection function on the calculator.

Say I have y=4xe^(-x^2) and y = |x|. How would I find this intersection.

Note. I really only need the x values because I am trying to then find the area between these two curves.

two cases ...

(1) for $x \geq 0$

$4xe^{-x^2} = x$

$4xe^{-x^2} - x = 0$

$x(4e^{-x^2} - 1) = 0$

$x = 0$

$e^{-x^2} = \frac{1}{4}$

$-x^2 = -\ln(4)$

since $x > 0$ ...

$x = \sqrt{\ln(4)}$

(2) for $x < 0$

$4xe^{-x^2} = -x$

$4xe^{-x^2} + x = 0$

$x(4e^{-x^2} + 1) = 0$

$x = 0$ only

seems I've solved this before ... ?
• Mar 14th 2009, 08:24 AM
gammaman
Thanks. One more question. I have not had to find intersections in a long time. What if I had something simple like

$x^3 = 3x+2$

$\Rightarrow x^3-3x-2=0$

seems I forgot how to find the roots when I have a power higher than 2.
• Mar 14th 2009, 10:31 AM
Quote:

Originally Posted by gammaman
Thanks. One more question. I have not had to find intersections in a long time. What if I had something simple like

$x^3 = 3x+2$

$\Rightarrow x^3-3x-2=0$

seems I forgot how to find the roots when I have a power higher than 2.

Use general way of factorising by trial and error and if you fail

• Mar 14th 2009, 12:07 PM
HallsofIvy
Quote:

Originally Posted by gammaman
Thanks. One more question. I have not had to find intersections in a long time. What if I had something simple like

$x^3 = 3x+2$

$\Rightarrow x^3-3x-2=0$

seems I forgot how to find the roots when I have a power higher than 2.

The "rational root" theorem says that any rational number satisfying this equation must be an integer that divides 2. That tells you that the only possible rational roots are 1, -1, 2, and -2. Try those to see if there is a rational root.

If not, then you will need to use "Ferrari's cubic formula" that ADARSH links to.