The general form of a equation of second degree is
ax2 + bx + c = 0
In this post we’ll deduce the general solution formula to that.
The constant c just shifts the curve vertically in the xy-plane.
Let’s start by determining the slope of the general case, which is given by the derivative of the general case: 2ax + b.
The zero point of the derivative gives the symmetry point of the general case. Let’s examine parabola x2 – 2x and its derivative 2x – 2:
The zero point of the derivative is at x = 1. We notice that the symmetry point of the parabola lies at this x-position, the roots of the parabola are at x = 0 and x = 2.
In general the symmetry point of equation of second degree lies at the zero point of its derivative.
Now we are ready to deduce the general solution formula.
Let us solve the zero point of the derivative: x = –b / 2a.
Next we put this in the general equation, and we get:
ax2 – b2 / 2a+ c = 0
By solving the x, we get
The formula isn’t ready yet, but we know how the derivative is related to this and how it behaves.
The zero point of the derivative behaves symmetrically to the previous, so all we have to do is subtract the solved x from the symmetry point and add the solved x to the symmetry point. We get:
That’s it! We have general solution formula of the equation of the second degree.
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I hope I haven’t chosen any funny words in this article. I don’t have my Finnish-Swedish-English math dictionary with me right now… 🙂