Two circles intersection

Could you tell what are all the four points in following? Two circles intersect at two points maximum when we want to draw intersecting circles. But there we are solving quadratic equations, what is the argument about the other two missing points ?

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The other two points aren’t missing. They’re located in the same exact place the first two points where.

It’s true that the procedure for solving a quadratic equation, $ax^2+bx+c=0$, yields two solutions,

$$x_\pm=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$$

But remember, sometimes you might have $b^2-4ac=0$, in which case the two solutions will be $x_+=\frac{-b+0}{2a}$ and $x_-=\frac{-b-0}{2a}$. This is called a repeated root since they equal each other.

This is basically what happens in the course of solving for the coordinates of intersection points of two circles. Technically, there are indeed four solutions; it’s just that two of them will coincide with the other two.