Given a real constant, \(c\), the equation \[x^4=(x-c)^2\] has four real solutions (including possible repeated roots) for

  1. \(c\leq \frac{1}{4}\),
  2. \(-\frac{1}{4} \leq c \leq \frac{1}{4}\),
  3. \(c \leq -\frac{1}{4}\),
  4. all values of \(c\).

How could we turn this question into a question about quadratics?

Remember that there are two possible solutions when we solve an equation of the form \(y^2=a\).

The blue curve is \(y = x^4 - (x-c)^2\).

What could we say about this curve when the above equation has “four real solutions”?