XXII.  Quadratic Equations

Quadratic Equations is simply a term to signify that we are working with equations where the variable has a power of 2.In other words, a quadratic equations looks like this:  I know, we just covered most of this in Factoring, but - think of this as a refresher.

ax2 + bx + c = 0

Where the a in front of the x squared is some constant, the b is some constant, and the c is some constant.  Note that a cannot equal zero (or it would get rid of the x squared term).  The formula might not start out looking like this, but if it can be configured to be in this form, it's quadratic.

Why do we call it "quadratic" why don't we just call it square?  I believe that it might be that, in the old days, mathematicians got picked on a lot.  So, they made up all these complex terms to sound intelligent (just one man's opinion).  Anyway, the word quad means 4.  If you want the area of a square (a four sided figure) you multiply a side by itself (you square the side).  you could look it up at dictionary.com, but they don't really give you the entomology either.

While we're talking about words.  The word coefficient means the constant being multiplied times a variable.  A constant is just a non-variable, like 3.

There are usually two values for the squared term (see the graph below, and notice most the values for y have two x values).

A graphed parabola.

Graphing Quadratic Equations

The graph of a quadratic is always a parabola (see figure above).  If the coefficient of the squared variable is positive, the curve will go up, if negative it will go down.

The Quadratic Formula

−b ± √   b2 − 4ac 

Remember in the first paragraph we showed you the formula for quadratics?  I'll repeat it.

ax2 + bx + c = 0

Every quadratic can be put in that form.  Now, look at the quadratic formula.  The b in the quadratic formula is the b in the above equation.  It is the coefficient of the variable.  The a is the coefficient of the variable squared, and the c is the coefficient of the constant.

This is a fairly complex formula, but it will give you both the solutions to every quadratic!  Notice it starts out (-b + or - the square root.... )  Well that is what gives you both answers.  There's a ( -b + the square root .... ) and a ( -b - the square root).

Factoring Quadratics

Often, it's easier just to factor the quadratic into it's multiples and solve each of the expressions.  Factoring is such a big topic - we've given it its own section - Factoring .  

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Quadratic Equations