A more important question: Why should there be infinitely many solutions?
The roots/solutions of a quadratic equation $ax^2 + bx + c = 0$ are given by looking at when the graphs $y = ax^2 + bx + c$ cuts the $x$-axis. Remember that $a$, $b$ and $c$ are all fixed numbers, e.g. $y = x^2 + 2x + 1.$ Ask yourself this: Why should there be infinity many values of $x$ for which $x^2 + 2x + 1 = 0$? In fact, there is only one single value of $x$ for which $x^2 + 2x + 1 = 0$. This is because $x^2 + 2x + 1 = (x+1)^2$ and so $x = -1$ is the only solution. Find me another value of $x$ for which $x^2 + 2x + 1 = 0$ and I will give you US1,000,000.
Draw yourself some sketches. Draw the one-parameter family of parabolae given by y = x^2 - k$ where $k$ is a number we're going to play with. When $k > 0$, the parabola cuts the $x$-axis at two distinct points: $x = \pm \sqrt{k}$. When $k = 0$, the parabola cuts the $x$-axis at one point: $x = 0$. When $k < 0$, the parabola misses the $x$-axis altogether.