SOLUTION: If the line y + 2x = p does not intersect the curve y^2 = x + p, find the range of values of p

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Question 1184258: If the line y + 2x = p does not intersect the curve y^2 = x + p, find the range of values of p
Found 2 solutions by Edwin McCravy, MathLover1:
Answer by Edwin McCravy(20063) About Me  (Show Source):
You can put this solution on YOUR website!
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%28y+%2B+2x+=+p%2Cy%5E2+=+x+%2B+p%29

Solve the first equation for y:

y=p-2x

Substitute in the second equation:

%28p-2x%29%5E2+=+x+%2B+p%29

p%5E2-4px%2B4x%5E2+=+x+%2B+p

4x%5E2-4px-x%2Bp%5E2-p=0

%284%29x%5E2%2B%28-4p-1%29x%2B%28p%5E2-p%29=0



x+=+%28%284p%2B1%29+%2B-+sqrt%28+%2816p%5E2%2B8p%2B1%29-16%28p%5E2-p%29+%29%29%2F8+

x+=+%28%284p%2B1%29+%2B-+sqrt%28+16p%5E2%2B8p%2B1-16p%5E2%2B16p+%29%29%2F8+

x+=+%28%284p%2B1%29+%2B-+sqrt%2824p%2B1%29%29%2F8+

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

Edwin

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

If the line y+%2B+2x+=+p+does not intersect the curve y%5E2+=+x+%2B+p, find the range of values of p
y+%2B+2x+=+p+.......eq.1
y%5E2+=+x+%2B+p..........eq.2
___________________________
y+=+p+-2x.......eq.1
y+=+sqrt%28p%2Bx%29..........eq.2

=> p+-2x=+sqrt%28p%2Bx%29.....square both sides
%28p+-2x%29%5E2=+%28sqrt%28p%2Bx%29%29%5E2
p%5E2+-+4px+%2B+4x%5E2=+p%2Bx
p%5E2+-+4px+%2B+4x%5E2-+p-x=0
4x%5E2-+4px-x%2Bp%5E2++-+p=0
4x%5E2-+%284p%2B1%29x%2B%28p%5E2++-+p%29=0
use discriminant
the line will not intersect the curve if
b%5E2-4ac+%3C++0
b%5E2-4ac+%3C0....a=4, b=-+%284p%2B1%29, c=%28p%5E2++-+p%29
%28-+%284p%2B1%29%29%5E2-4%2A4%28p%5E2++-+p%29+%3C0
16p%5E2+%2B+8+p+%2B+1-16p%5E2++%2B16+p+%3C++0++
8p+%2B+1+++%2B16p+%3C++0
24p+++%2B1%3C0
p+%3C++-1%2F24+
the line will not intersect the curve if p+%3C++-1%2F24+