Solving Linear-Quadratic Systems

Question

If a quadratic function is represented by y= kx^2-5x+1 and a linear system is represented by y=4x+k, find the value of k where the two functions will only have one point of intersection.

Answer 1

To say that the two curves have only one point of intersection is to say that their difference has only one root.

Their difference, kx²-9x+1-k=0, is a quadratic when k≠0, so it will have one root exactly when the discriminant is 0.

(-9)²-4(k)(1-k)=0

81-4k+4k²=0

But the discriminant of this quadratic is 16-4(4)(81), which is less than 0, so the only nonzero k's that give a unique solution are nonreal.

So it follows that k must be 0. When this is the case, your equations are just straight lines, which of course have only one intersection point.