Question 1184258
<pre>
We solve them simultaneously.  If the solutions are real, they will be the
values at which they intersect, and if they are imaginary they will not
intersect:

{{{system(y + 2x = p,y^2 = x + p)}}}

Solve the first equation for y:

{{{y=p-2x}}}

Substitute in the second equation:

{{{(p-2x)^2 = x + p)}}}

{{{p^2-4px+4x^2 = x + p}}}

{{{4x^2-4px-x+p^2-p=0}}}

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

{{{x = (-(-4p-1) +- sqrt( (-4p-1)^2-4*(4)*(p^2-p) ))/(2*(4)) }}}

{{{x = ((4p+1) +- sqrt( (16p^2+8p+1)-16(p^2-p) ))/8 }}}

{{{x = ((4p+1) +- sqrt( 16p^2+8p+1-16p^2+16p ))/8 }}}

{{{x = ((4p+1) +- sqrt(24p+1))/8 }}}

That will not intersect when 24p+1 < 0, or when p < -1/24

Edwin</pre>