Transcript Solving Quadratic Equations by Completing the Square

Completing the Square
Perfect Square Trinomials
Examples
2
  x  3
2
 x + 6x + 9
2
 x2 - 10x + 25   x  5 
2
2
 x + 12x + 36   x  6 

Creating a Perfect
Square Trinomial
In the following perfect square
trinomial, the constant term is
missing.
X2 + 14x + ____
 Find the constant term by
squaring half the coefficient of
the linear term.
 (14/2)2
X2 + 14x + 49

Perfect Square Trinomials
Create perfect
square trinomials.
 x2 + 20x + 100
___
4
 x2 - 4x + ___
 x2 + 5x + 25/4
___

Quadratic Functions
2
f ( x)  ax  bx  c
GENERAL form:
a0
If a>0 it opens UP
If a<0 it opens DOWN
STANDARD or VERTEX
form:
2x2 + 3x - 5
f ( x)  a( x  h)  k
( h, k )
where
vertex.
is the
2
f ( x)  2( x  3)2  1
Changing to Standard Form
by “Completing the Square”
Step 1: Group f(x) = x2 + 8x + 11
the 1st 2 terms f(x) = (x2 + 8x) + 11
Step 2: Add & subtract blanks
2
42
f(x) = (x2 + 8x + 4_)
+ 11 - _
Step 3: ½ the middle term squared
2 + 11 - 16
f(x)
=
(x
+
4)
Step 4: factor
Step 5: simplify f(x) = (x + 4)2 - 5
f(x) = 2x2 + 8x + 7
Step 1: Group the 1st 2 terms
f(x) = (2x2 + 8x) + 7
Step 2: Factor out the 2
f(x) = 2(x2 + 4x) + 7
2
2 + 7 - 2_(2)
f(x) = 2(x2 + 4x + 2_)
f(x) = 2(x + 2)2 + 7 - 8
f(x) = 2(x + 2)2 - 1
f(x) = -x2 - 4x + 21
YOUR TURN
2 + 21 - 2
2
f(x) = -(x2 + 4x + 2_)
_(-1)
f(x) = -(x + 2)2 + 21 + 4
f(x) = -(x + 2)2 + 25