+367 For what negative value of \(k\) is there exactly one solution to the system of equations \(y=2x^2+kx+6, y=-x+4\)?
+33659 We need the two functions to be equal, and their slopes to be equal.
\(2x^2+kx+6=-x+4\\4x+k=-1\\ \text{so:}\\2x^2+(k+1)x+2=0\\4x+(k+1)=0\)
Use the second to replace k+1 in the first and solve for x
\(2x^2-4x^2+2=0\\2x^2 = 2\\x^2=1\\x_1=1\\x_2=-1\)
Now \(k=-1-4x\) so I'll leave it to you to decide which value of x to substitute in here in order to obtain a negative value for k.
