SOLUTION: For what value(s)of k will the graph of y=3x^2+(k+2)x+3 touch the x-axis at exactly one point

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Question 184671: For what value(s)of k will the graph of y=3x^2+(k+2)x+3 touch the x-axis at exactly one point
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
y+=+3x%5E2+%2B+%28k%2B2%29x+%2B+3
Instead of 2 roots, r%5B1%5D and r%5B2%5D,
There will be a double root, r
%28x+-+r%29%28x+-+r%29+=+0
x%5E2+-+2rx+%2B+r%5E2+=+0
-------------------------
3x%5E2+%2B+%28k%2B2%29x+%2B+3+=+0
x%5E2+%2B+%28%28k%2B2%29%2F3%29%2Ax+%2B+1+=+0
I can make the substitutions
%28k%2B2%29%2F3+=+-2r
r%5E2+=+1
r+=+1
and
r+=+-1
-------------------------
%28k%2B2%29%2F3+=+-2%2A1
k+%2B+2+=+-6
k+=+-8
and
%28k%2B2%29%2F3+=+-2%2A%28-1%29
k+%2B+2+=+6
k+=+4
The graph touches the x-axis when k+=+-8 or k+=+4
for k+=+-8:
y+=+3x%5E2+%2B+%28k%2B2%29x+%2B+3
y+=+3x%5E2+%2B+%28-8%2B2%29x+%2B+3
y+=+3x%5E2+-+6x+%2B+3
for k+=+4:
y+=+3x%5E2+%2B+%284%2B2%29x+%2B+3
y+=+3x%5E2+%2B+6x+%2B+3
I'll plot both:
+graph%28+500%2C+500%2C+-5%2C+5%2C+-2%2C+8%2C+3x%5E2+-+6x+%2B+3%2C3x%5E2+%2B+6x+%2B+3%29+