Calculating the intersection

At an intersection, two function graphs intersect.

Example

Calculate the intersection / intersections of the graphs of $f(x)=-(x+2)^2+1$ and $g(x)=(x+2)^2-1$

1. Equate function equations

   
   $f(x)=g(x)$
   $-(x+2)^2+1=(x+2)^2-1$



2. Change equation to zero

   
   $-(x^2+4x+4)+1= x^2+4x+4-1$
   $-x^2-4x-4+1= x^2+4x+4-1$
   $-x^2-4x-3=x^2+4x+3\quad|+x^2$
   $-4x-3= 2x^2+4x+3\quad|+4x$
   $-3= 2x^2+8x+3\quad|+3$
   $0=2x^2+8x+6$

3. Bring the equation into the reduced quadratic equation

   
   $0=2x^2+8x+6\quad|:2$
   $0=x^2+\color{green}{4}x+\color{blue}{3}$

4. Apply quadratic formula

   
   $x_{1,2} = -\frac{\color{green}{p}}{2} \pm\sqrt{(\frac{\color{green}{p}}{2})^2-\color{blue}{q}}$
   $x_{1,2} = -2\pm\sqrt{4-3}$
   $x_{1,2} = -2\pm\sqrt{1}$
   $x_{1,2} = -2\pm1$

   $x_{1} = -2+1=-1$
   $x_{2} = -2-1=-3$

5. Insert $x_{1}$ and $x_{2}$ in one of the two equations

   
   $g(-1)=(-1+2)^2-1=0$
   $g(-3)=(-3+2)^2-1=0$



6. Two intersections:

   $S_{1}(-1|0)$ and $S_{2}(-3|0)$