# Thread: find intersections without calculator

1. ## 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.

2. 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 $\displaystyle x \geq 0$

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

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

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

$\displaystyle x = 0$

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

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

since $\displaystyle x > 0$ ...

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

(2) for $\displaystyle x < 0$

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

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

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

$\displaystyle x = 0$ only

seems I've solved this before ... ?

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

$\displaystyle x^3 = 3x+2$

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

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

4. 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

$\displaystyle x^3 = 3x+2$

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

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

5. 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

$\displaystyle x^3 = 3x+2$

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

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.