Discovering THE QUADRATIC FORMULA

We started today's class with this simple question. .

FIND THE ROOTS USING 'COMPLETING THE SQUARE':
i) y=x²-6x-3
y=x²-6x+9-9-3
y=(x-3)²-12
* * Okay so first of all, we simply transfored our original equation from general form to standard form.

- NOW WE MUST ALLOW Y = 0
0=(x-3)²-12
12=(x-3)²
+-(12)^1/2=x-3
3+-2(3)^1/2=x

ii) y=2x²-8x+1
y=2[x²-4x+4-4]-1
y=2[(x-2)²-4]+1
y=2(x-2)²-7

0=2(x-2)²-7
7=2(x-2)²
7/2=(x-2)²
[7/2]^1/2=x-2
2[7/2]^1/2=x

Is there an EASIER APPROACH to find roots of all parabolas of the general form?
- YES. Yes there is!

Quick Note: y=ax²+bx+c
- represents ALL, yes every single parabola out there


CONVERTING THE GENERAL FORM TO THE STANDARD FORM FOR THE EQUATION OF A PARABOLA:

i) The General Form
y=ax²+bx+c

ii) Factor "a" out of all x-terms
y=a[x²+b/ax ]+c

iii) Find half the coefficient of the x-term & square it
1/2(b/a)=b/2a (b/2a)² = b²/4a²

iv) Complete the square
y=a[x²+b/ax+(b/2a)²-(b/2a)²]+c

v) Factor and Simplify
y=a[(x+b/2a)²-b/4a²]+c

vi) Distribute "a". Find common denominator
y=a(x+b/2a)²-b²/4a+(4a/4a)c

vii) Add Fractions
y=a(x+b/2a)²+-b²+4ac/4a

IN CONCLUSION..
Reference: y=a(x-h)²+k

h= -b/2a
**REMEMBER THAT!

k= -b²+4ac/4a
**YEAH, THATS UGLY


FINDING THE ROOTS OF A PARABOLA FROM THE EQUATION IN GENERAL FORM CONTINUED FROM ABOVE:

i) Set to "zero"
0=a(x+b/2a)² + -b²+4ac/4a

ii) Add b²+4ac/4a to both sides
b²+4ac/4a=a(x+b/2a)²

iii) Mulitiply both sides by 1/a

b²-4ac/4a²=(x+b/2a)²

iv) Find the square root of both sides
[b²-4ac/4a²]^1/2 = x+b/2a

v) Simplify Radical
-b/2a+-[b²-4ac/2a]^1/2=x+b/2a

vi) Add -b/2a to both sides
-b/2a+-[b²-4ac/2a]^1/2=x

IN CONCLUSION...
vii) We add the fractions and get
(dun dun dun..) THE QUADRATIC FORMULA!

x= -b+-[b²-4ac]^(1/2) / 2a

REMEMINDER: It is important to remember this specific formula!

So for the finale of my scribe- heres our handy sing along.
(8).. X equals negative b,
plus or minus the square root of b square,
minus 4ac, all over 2a
.

.

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REMINDER: In order to transfer information from short term memory into long term memory, take in the information SIX TIMES.

Okay, my times up guys. Hope you enjoyed
- Homework assignment: Exercise #14

sandy you're next!

- - Nightt :)