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Beginning students often become confused about the difference between the quadratic function, the quadratic equation, and the quadratic formula. They all sound a little strange at first, and a little similar.

Let’s look at the quadratic function first: ƒ(x) = ax2 + bx +c

It’s just a shorthand way of saying you can make a shape (in this case a parabola) by plugging in all the values of x. Pick any number on the x axis, plug it into the function, and put a dot above or below it at the height that matches your answer for ƒ(x). On a three day weekend you could plot enough points to discover that they all connect together in a smooth line we call a parabola.

Now let’s consider the quadratic equation: 0 = ax2 + bx +c

All it is is that one special case of the quadratic function when ƒ(x) = 0. I just cut and pasted the right hand side to get it down here. Turns out it’s pretty useful but I won’t go into that here.

And now the quadratic formula: X = -b ± v(b2-4ac)
2a

Centuries ago, some guy a lot smarter than me looked at the quadratic equation and said, “Hey, wouldn’t it be handy if we had a way to figure out what x was anytime we were given a, b, and c?” Then he figured it out, and that’s what the quadratic formula is.

So the takeaway is:

quadratic function: ƒ(x) = ax2 + bx +c

quadratic equation: 0 = ax2 + bx +c

quadratic formula: X = -b ± v(b2-4ac)
2a