Factoring 2 - Completing the Square
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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:
β’ π¦ = β4.9π₯ 2 + 50π₯ + 150
β’ We can look at this equation and determine a few things
about the Projectile Motion Function.
β’ π¦ = ππ₯ 2 + π£0 x + π 0
β’ x=time
β’ y=height
β’ a= downward acceleration due to gravity:
-4.9 m/s 2 or -16 ft/π 2
β’ π£0 = initial upward velocity
β’ π 0 =initial height in meters or feet.
Follow up
β’ Now find the maximum height reached by the
rocket, and how many seconds it took to get
there.
β’ In order to answer this question we need to
know the______?
β’ Can we determine the vertex given the
general equation?
β’ What form to we need?
Goal
β’ To convert a quadratic equation from General
Form to Vertex Form.
β’ Method: Completing the Square.
Factoring: Completing the Square
General to Vertex form
β’ We want to go from
ππ₯ 2 + ππ₯ + π
To
π π₯ββ
2
+π
Completing the Square Method
The two relationships that will allow us to
make this conversion are:
β’ β=
π
β
2π
and
β’ π=πβ
π2
4π
Example 1
β’ Write π = ππ β πππ + ππ in vertex form.
β’ Step 1: identify a,b and c.
β’ a=1, b=-10, c=15
Contβ¦
β’ Step 2: substitute values into the relations β =
π
π2
β
and π = π β
2π
β’ β=
4π
β10
β
2β1
β’ π = 15 β
β 10
=
10
2
β10 2
4β1
=5
= 15 β
100
4
= 15 β 25 =
Contβ¦
β’ Step 3: Substitute the values for h and k into
the vertex form.
β’ Vertex form: π π₯ β β 2 + π
β’ h=5, k=-10, a=1 (never changed)
β’ Solution: π₯ β 5
2
β 10
Example 2
β’ Factor π₯ 2 + 10π₯ + 25
β’ a=_____, b=______, c=_______
β’ β=
π
β
2π
and π = π β
π2
4π
Find h and k.
β’ Plug h and k into π π₯ β β 2 + π
β’ Answer: x + 5 2 + 0 = x + 5 2
State the vertex
β’ What is the vertex of the previous problem?
x+5 2
β’ Answer: (-5,0)
Example 3: You try
β’ What is the vertex of π¦ = π₯ 2 β 6π₯ + 11?
Example 4: You try again
β’ Write the following equation in vertex form.
β’ π¦ = 3π₯ 2 β 12π₯ + 18
Example 5
β’ Write the following equation in vertex form.
β’ π¦ = (π₯ β 3)(π₯ β 9)
β’ Hint: first convert factored form to general
form, the change to vertex form)
Example 6
β’ Find the vertex:
β’ π¦ = β4(π₯ + 1)(π₯ + 3)
Homework
β’ 7.3: Skip #2
β’ This is a big assignment. Pace yourself!
β’ The only way to really understand this stuff is
to Practiceβ¦A LOT!