The Quadratic Formula Explained

By Cias Hart
Quadratic equations are an important stepping stone in algebra, geometry, and calculus. Quadratic formulas are simply polynomial equations of the second degree: ax^2+bx+c=0. This represents a graph of a concave shape which graphs formulas representing exponential growth and decay, maximizing or minimizing at a point.

While much of the theory behind a quadratic equation is solely for those reaching into the upper levels of calculus, it is still important to know how to solve one as early as the freshman year of high school. Below, I'll describe each component of the quadratic formula and how to apply it.

If we toy with the quadratic formula, we start to see the problem in solving it. We end up with x on each side of the equation, or we end up trying to take the root of a complex problem. We could delve into how to reach the final "x=" stage, but it has been well established that the following is the correct quadratic formula:

x = (-b +/- Sq (b^2 + 4ac ) ) / 2a

Step by Step

For our purposes, we'll use 4x^2+8x+12=0 as our formula.

First Things First, The Top

Step One: The "b" Variable

First, identify "b." This variable is the second coefficient in the quadratic formula. In our case, it's 8. The first thing we'll plug in will be this. This leaves us with ( -8 +/- Sq (8^2 + 4ac ). 8^2 is 64, so we can further simplify ( -8 +/- Sq (64+4ac) ).

Step Two: The "c" Variable

Next, we will need to plug in the "c" coefficient from our original formula. This value is 12, since c represents the coefficient without an x next to it. So, our formula is now ( -8 +/- Sq (64+(4)a(12) ). We further simplify to get ( -8 +/- Sq (64+48a) ).

Step Three: The "a" Variable

Now, let's plug in our last value, "a."

( -8 +/- Sq (64+48(4)) )

Becomes:
( -8 +/- Sq (256 ) )

In words, negative eight plus or minus the square root of two hundred and fifty six.

This is the top of our equation.

Next, the bottom

This part is simple. Plug in "a," in our case 4, and evaluate:

2 (4)
2*4=8
That's it!

Now, put the whole equation together:

(-8 +/- Sq (256 ) ) / (8)

We can simplify the square root of 256 to be 16.

( -8 +/- 16 ) /8

x, therefore equals either

( -8 + 16 ) / 8, or 1
or
(-8 - 16 ) /8 or -3

x = 1 or -3

That's it! Because the physical nature of a quadratic equation is a concave, it will have two different x values, meaning it will cross the x axis twice.

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