Quadratics (Completing the Square and our Wonderful Quadratic Equation)

Completing the Square:

I've included a link if you're wanting to go back and remember how completing the square works.  Remember, we just want to make the right side of the equation a perfect square trinomial (which is in the form of ax^2 + 2kx + k^2 --- don't worry too much about the k's.  It's just to show you that your b value for the quadratic is just 2k and your c value for the quadratic is just k^2.

The idea behind Completing the Square

Steps:
1.  get the equation so that you have x^2 + 2kx   on the right, and = - c on the left.
2.  Find your b term
3.  Divide your b term by 2
4.  Got that answer?  Square it.
5.  That's now your new c value (which we can also label as k^2).

Once you have that new c value, you factor the trinomial into it's perfect square factored form, take the square root of both sides of your equation, and solve for x (remember, that's your root!)  Don't forget when we take the square root of a number, it's going to be either that number, or the negative of that number.  ((-4)(-4) = 16     OR  (4)(4) = 16).

EXAMPLES!

x² – 2x – 48 = 0

x² – 2x – 48 + 48 = 0 + 48

x² – 2x         = 48

Our b term is -2.  Half of -2 is -1, then we square that, and get (-1)² = 1

x² – 2x + 1 = 48 + 1

x² – 2x + 1 = 49

(x + 1)(x + 1) = 49

(x + 1)² = 49

sqrt((x + 1)²⁾ = +/- sqrt(49)

(x + 1) = +/- sqrt(49)

So we have two situations:  either….  

(x + 1) = + 7   OR   (x + 1) = - 7

         x = 6      OR            x = -8

Quadratic Formula:

Remember, it might look scary, but this thing is going to be your new best friend.

Our general quadratic formula will be in the form:  

ax² + bx + c = 0

The a, b, and c values from your equation just replace their respective letters in the formula.  

Example:

x²  - 2x - 1 = 0

a = 1

b = -2

c = -1

x = -(-2) +/- sqrt( (-2^2) - 4(1)(-1))  / 2(1)

(Sorry, blogger doesn't like it when I try and type these equations to look beautiful in word and copy them into here - sqrt( ) is just the square root of whatever is in the parentheses!)

x = 2 +/- sqrt( 4 + 4 ) /  2

x = 2 + sqrt(8) / 2   OR  x = 2 - sqrt(8) / 2

(Aside:  We can break 8 into 2*4, and 4 is a perfect square, so we can take the square root of the 4 and leave the left over 2 under the square root, so it'll look something like, 2sqrt(2))

Since we can pull a factor of 2 out of both terms in the numerator (with the little amount of math magic / root properties), we find:

x = 2(1 + sqrt(2) / 2   OR  x = 2(1 - sqrt(2)) / 2

The twos in the numerator and denominator cancel out, leaving us with:

x = 1 + sqrt(2)    OR  x = 1 - sqrt(2)

TADA!  

Links!

Completing the Square Video

Quadratic Formula Video  

NOTE:  I really did record myself singing the quadratic formula song, but it's not uploading at the moment.  I will figure it out, though!