When does $x^4=(x-c)^2$ have four real roots?

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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 \(y^2=a\).

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