Solve: 3x-squared plus x equals 14
I'll remember when we're solving a quadratic equation
what we want to do is get that quadratic equation
into standard form and the standard form has this
ax-squared plus bx plus c equals 0
so everything equals 0.  So let's do that right away
let's subtract 14 from each side so we're left with 0 on the right hand side
I'll show that.  And then write the result, and that gives us
3x-squared plus x minus 14 equals 0
Notice the standard form is ax-squared, 
well our a then would be equal to 3
and our b,
that is the coefficient of x, is 1.
If there's nothing there you can think of it as 1 times x
so b equals 1.
Our c term
the number is -14.  You can think of this as a plus -14
okay so c equals -14
Now, we're going to use the quadratic formula
Let's write that down Just so you can
Note that quadratic formula
and
This is kind of nice, for something like this, it doesn't factor real easily
You could complete the square you'd have to divide by 3
but
let's try this
Quadratic Formula and that is
x equals -b plus or minus the square root of
b-squared minus 4 times a times c
all over 2a.
Make sure that you put that fraction bar under the -b as well.
There's a little song and I can't even say it with a straight face because it's kind of silly
but it'll help you remember
*singing*
x Equals negative B plus or Minus the Square root, of B Squared Minus 4ac all over 2a
Okay, so, we have our a and b and c and we just plug it into that quadratic formula.  Let's do that.
So i'm going to draw a line here and start this out
x Equals the opposite of b so -1
plus or minus the square root of b-squared
In this case that's 1.  But 1-Squared
minus 4 times a times 3
times c for ac and c is -14
all over 2 times a which again a is 3
Now, we can start to simplify this so x equals
-1 plus or minus, it's very important that you put the plus or the minus.
the square root of 1 and let's see... 
 -4 times 3 times -14
negative times a negative makes a positive.
So we'll say plus,
and the product of those three numbers is 168
all over 2 times 3 is 6
Continuing with that x equals -1 plus or minus the square root of
169
all over 6
Let's take this
up here and we'll continue this.
x then equals
-1
plus or minus, the square root of 169 is 13
all over 6.
Okay, now, we have two distinct solutions
and if you're just making
a note to yourself, this is okay
But typically you're turning something in or giving an answer, and so let's show those two different solutions
okay -1 plus 13 over 6
or, because it's plus or minus
-1 minus 13 over 6
so -1 plus 13, that's 12 over 6, and that's 2
We also have -1 minus 13 that's -14
6 and that equals -7 over 3
once you simplify that
So way down at the bottom I'm going to write this
solution set that Is
-7 over 3
typically writing the lower number first
and 2.
So these are both solutions to the original Quadratic Equation
