Transcript Factoring 2 - Completing the Square

Matching Puzzle
Grab a pair of scissors, a
puzzle sheet and sit at your
desk.
Warm up: Write the WHOLE
problem down.
1. The data below represents the height of a
rocket shot from 150 ft above ground and
traveling at a velocity of 50 m/second. Find
the quadratic equation that models this data.
Seconds 0
Meters 150
1
195.1
2
3
230.44 255.9
4
271.6
5
277.5
2. What form of the quadratic equation did we
get from this data?
Answer:
• 𝑦 = −4.9𝑥 2 + 50𝑥 + 150
• We can look at this equation and determine a few things
about the Projectile Motion Function.
• 𝑦 = 𝑎𝑥 2 + 𝑣0 x + 𝑠0
• x=time
• y=height
• a= downward acceleration due to gravity:
-4.9 m/s 2 or -16 ft/𝑠 2
• 𝑣0 = initial upward velocity
• 𝑠0 =initial height in meters or feet.
Follow up
• Now find the maximum height reached by the
rocket, and how many seconds it took to get
there.
• In order to answer this question we need to
know the______?
• Can we determine the vertex given the
general equation?
• What form to we need?
Goal
• To convert a quadratic equation from General
Form to Vertex Form.
• Method: Completing the Square.
Factoring: Completing the Square
General to Vertex form
• We want to go from
𝑎𝑥 2 + 𝑏𝑥 + 𝑐
To
𝑎 𝑥−ℎ
2
+𝑘
Completing the Square Method
The two relationships that will allow us to
make this conversion are:
• ℎ=
𝑏
−
2𝑎
and
• 𝑘=𝑐−
𝑏2
4𝑎
Example 1
• Write 𝒚 = 𝒙𝟐 − 𝟏𝟎𝒙 + 𝟏𝟓 in vertex form.
• Step 1: identify a,b and c.
• a=1, b=-10, c=15
Cont…
• Step 2: substitute values into the relations ℎ =
𝑏
𝑏2
−
and 𝑘 = 𝑐 −
2𝑎
• ℎ=
4𝑎
−10
−
2∙1
• 𝑘 = 15 −
− 10
=
10
2
−10 2
4∙1
=5
= 15 −
100
4
= 15 − 25 =
Cont…
• Step 3: Substitute the values for h and k into
the vertex form.
• Vertex form: 𝑎 𝑥 − ℎ 2 + 𝑘
• h=5, k=-10, a=1 (never changed)
• Solution: 𝑥 − 5
2
− 10
Example 2
• Factor 𝑥 2 + 10𝑥 + 25
• a=_____, b=______, c=_______
• ℎ=
𝑏
−
2𝑎
and 𝑘 = 𝑐 −
𝑏2
4𝑎
Find h and k.
• Plug h and k into 𝑎 𝑥 − ℎ 2 + 𝑘
• Answer: x + 5 2 + 0 = x + 5 2
State the vertex
• What is the vertex of the previous problem?
x+5 2
• Answer: (-5,0)
Example 3: You try
• What is the vertex of 𝑦 = 𝑥 2 − 6𝑥 + 11?
Example 4: You try again
• Write the following equation in vertex form.
• 𝑦 = 3𝑥 2 − 12𝑥 + 18
Example 5
• Write the following equation in vertex form.
• 𝑦 = (𝑥 − 3)(𝑥 − 9)
• Hint: first convert factored form to general
form, the change to vertex form)
Example 6
• Find the vertex:
• 𝑦 = −4(𝑥 + 1)(𝑥 + 3)
Homework
• 7.3: Skip #2
• This is a big assignment. Pace yourself!
• The only way to really understand this stuff is
to Practice…A LOT!