__More Solving Quadratic Equations:
The Quadratic Formula__

Again, what we've all been waiting for.

The quadratic formula is pretty easy to remember, and will always work, even if the solutions to the quadratic equation aren't always real numbers (they can be imaginary - the square roots of negative numbers). So here it is:

**
_______
x = -b ±
√b^{2} - 4ac
2a
**

First, just try it. Then we will do some more work with
it. Take the equation 2x^{2} + 7x + 6 = 0.

__________

x = __-7±√7 ^{2}
- 4(2)(6)
__
2(2)

x = __-7 ± ____√1
__ 4

**x =** -8/4 = **-2**
**OR**
**x =** -6/4 = **-1.5**

Try x^{2} - 2x + 4 = 0.

___________

x = __-(-2)±√(-2) ^{2} - 4(1)(4)
__
2(1)

___

x = __4±____√-12
__ 2

As you can see, this equation has no real solutions, because we need to take the square root of a negative number.

Now that you have seen these two examples, it is time to see how the quadratic formula works, and it works on the basis of the "completing the square" process.

So we have the standard form of the
equation: ax^{2} + bx + c = 0

Divide each term by a, to get x^{2}
alone (you need this for completing the square):

x^{2} + __ b __x + __ c __
= 0

a a

Then move the constant, c/a, to the other side by subtracting it
from both sides:

x^{2} + __ b __x = -__ c
__
a a

Now on the left side we want to complete the square. Remember
that that form is a^{2} + 2ab + b^{2}. Basically, in
completing the square, to get the term of b^{2}, when the coefficient of
x^{2} is 1, we just add the square of half the coefficient of x.
Ok...in a simpler showing:

__ 1 __ * __ b __ = __
b __

2 a 2a

__(____ b ^{2}__ =

2a) 4a

Add this ^^ to both sides.

x^{2} + __ b __x + __
b ^{2} __ = -

a 4a

We factor the left side and rearrange the right side.

(x + __ b )__^{2} = __
b ^{2} __ -

Now get a common denominator for the right side. It will
be 4a^{2}.

(x + __ b )__^{2} = __
b ^{2} __ -

(x + __ b )__^{2} = __
b ^{2} - 4ac __

Take the square root of each side, but remember on the right
side to put plus OR minus.

_________

x + __ b __ = ±__√____
b ^{2} - 4ac __

(I can't show it well, but on the right side, it's the square
root of the WHOLE SIDE, not just the top line. Now simplify the right side
again.

________

x + __ b __ = ±__√____
b ^{2} - 4ac __

________

x +

We're almost done! Subtract b/2a from both sides, to move
it to the right.

________

x = - __ b __ ± __√____
b ^{2} - 4ac __

Here's the last step: Simplify the right side *again*.

________

x = - __ b ± ____√____
b ^{2} - 4ac __

Recognize this??? Well you should...*gives people who don't glares*...just kidding. It really takes awhile to get the hang of all this. You may have to go over this explanation more than a few times to get it. You should know that it is easiest to factor a quadratic, if possible, but this can also be used. Happy quadratic equation solving!!

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