The Quadratic Equation Formula: The meaning of the term -b/2a


You have probably wondered many times what is the meaning of the quadratic formula. I mean, you probably know how to use the formula, this is, if you are presented with a problem involving some kind of quadratic equation, you know that the following formula needs to be used:

\[x = \frac{-b \pm \sqrt{b^2-4ac} }{2a}\]

For example, if you are asked to solve the equation: \(2x^2 -10x + 12 = 0\), then you know it is a quadratic equation, and in this case, \(a = 2\), \(b = -10\) and \(c = 12\). So then we have to plug those values into the quadratic equation formula:

\[x = \frac{-b \pm \sqrt{b^2-4ac} }{2a} = \frac{-(-10) \pm \sqrt{(10)^2-4(2)(12)} }{2(2)}\] \[= \frac{10 \pm \sqrt{100-96} }{4} = \frac{10 \pm \sqrt{4} }{4} = \frac{10 \pm 2}{4}\]

which means that the solutions are \(x_1 = 2\) and \(x_2 = 3\).

But what is the meaning of the term -b/2a in the quadratic formula?? It is very useful to have the proper intuition about it.

The term -b/2a has a clear graphical interpretation, and it corresponds to the position of the symmetry axis that is defined by the graph of the quadratic formula. So then, simply, the term -b/2a is the “center” of the parabola defined by a quadratic equation.

You can see a video below with a good tutorial on how to use the quadratic equation in various different contexts.




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