Quadratic functions - nature of roots

Find the range of values of k for which the quadratic equation

2x² – 2x + k + 2 = kx has real roots.

This is a classic question that makes use of the determinant of the quadratic equation. The determinant of a quadratic equation is D = b² – 4ac.

When D = 0, then it means that the equation has equal roots; i.e. the two answers of x for the quadratic equation are the same.

When D > 0, the roots will be real and distinct; i.e. there will be two different answers for x.

When D < 0, then the roots will be imaginary. This means that there are no answers to the quadratic equation and graphically, it means that the curve does not intersect the x axis at all.

Now, what about when the roots are simply real?  Then it means that D >= 0.  So, apply this to our question, we will first rearrange the equation so that it will equal to zero. Hence,

2x² – 2x  - kx + k + 2 = 0

or

2x² – (2 + k)x + (k + 2) = 0

For real roots,

Since this is a positive k² function, then