# Thread: Find intersecting coordinates of two functions

1. ## Find intersecting coordinates of two functions

find the x-coordinates of the points of intersection for the following functions below

$\\f(x) = x^{2} + 12x + 32\\g(x) = -x^{2} - 12x - 31$

My calculator shows 2 parabolas that cross, but the answers have square roots in them (showing me that it should be done w/o a calculator). Can anyone show me how to find the intersection of a parabola? Thanks!

2. ## Re: Find intersecting coordinates of two functions

Originally Posted by sykotic95
find the x-coordinates of the points of intersection for the following functions below

f(x) = (x^2) + (12x) + (32)
g(x) = (-x^2) - (12x) - (31)

My calculator shows 2 parabolas that cross, but the answers have square roots in them (showing me that it should be done w/o a calculator). Can anyone show me how to find the intersection of a parabola? Thanks!
As the coefficients of the squared term have opposite signs,
one of these is a U-shaped parabola and the other is an inverted U-shape.

They touch where

$x^2+12x+32=-x^2-12x-31\Rightarrow\ x^2+12x+32-\left(-x^2-12x-31\right)=0$

This solutions of this equation (where a new parabola crosses the x-axis)
are also the solution to where the parabolas intersect.

You may use the quadratic equation to solve.

3. ## Re: Find intersecting coordinates of two functions

Wow. That was so obvious. I'll try it now and edit this post with an update when I have the answer. Thanks!

Edit:
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$\\x^{2}+12x+32=-x^{2}-12x-31\\\\x^{2}+12x+32-(-x^{2}-12x-31)=0\\\\2x^{2}+24x+63=0\\\\x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\\\\x=\frac{-24\pm\sqrt{24^{2}-4(2)(63)}}{2(2)}\\\\x=\frac{-24\pm\sqrt{576-504}}{4}\\\\x=\frac{-24\pm\sqrt{72}}{4}\\\\x=-6\pm\frac{1}{4}\sqrt{72}\\\\x=-6\pm(\frac{1}{4})(2)(3)\sqrt{2}\\\\x=-6\pm\frac{3}{2}\sqrt{2}$