1.
Today:
Warm-Up: Number Sense
Warm-Up: Review Quadratic Graphs
Introduction to Completing the Square
Class Work
1
2.
Number Sense: Time
It is 12:00 noon on a February Friday in
Saipan. What time and day is it in...
1. Tokyo, Japan
4. Honolulu, Hawaii3. South Pole, Antarctica
2. New York, NY
1. Friday, 11:00 am
4. Thursday, 4:00 pm3. Friday, 3:00 pm
2. Friday, 3:00 am
3.
4. 90% of 90 girls and 80%
of 110 boys have shown up
at the stadium on time.
How many people are late?
Warm-Up:
3
4.
Warm-Up:
5. If the parabola y = x2 is flipped upside down, made 5
times as wide, and shifted 8 units down the y-axis,
write the equation for the new parabola.
6. If the parabola y = x2 is flipped upside down, made
twice as narrow, and shifted 6 units up the y-axis, write
the equation for the new parabola.
y = -⅕x2 - 8
y = -2x2 + 6
4
10.
10
Completing the Square
Factoring “unfactorable” 2nd degree trinomials
11.
11
• We have learned earlier that a perfect square trinomial
can always be factored.
• Therefore, if we have a trinomial we cannot factor
using integers, we can change it in such a way that we
are dealing with a perfect square trinomial.
Completing the Square:
12.
12
• Recall that a perfect square trinomial is always in the
form:
• Therefore, we have to change the polynomial so that it
fits the form.
• To really learn this, go through each step of the
process. Your goal should be to learn the steps in order.
22
2 baba
Completing the Square:
13.
13
The equation we are going to solve is the following…
By testing whether or not the factors of c can sum to equal
b, we can determine if the trinomial is factorable. This
trinomial is not factorable in its present form. However,
with our new tool, we can solve this previously
'unsolvable' quadratic.
2
2 16 20 0x x
There are five steps in this process, let's write them
down.
Completing the Square:
14.
Step 1
Divide by the leading coefficient
to set the a-value to 1.
14
2
2
2 16 20 0
2
8 10 0
x x
x x
Completing the Square:
15.
Step 2 Re-write the equation
in the form ax + by = c
15
2
2
2
8 10 0
8 10 10 10
18 0
0
x x
x x
x x
Completing the Square:
16.
Step 3 Find one-half of the b value.
Add the square of that number to
both sides.
16
2
2
2
2 2
8 10
8 104 4
16 28 6
x x
x x
x x
2
2
b
Completing the Square:
17.
Step 4
A) Re-write the perfect square
trinomial as a binomial squared.
B) Find the square root of each side
of the equation.
17
2
2
4
8 16 26
26
264
x
x
x x
Completing the Square:
18.
Step 5 Solve for x.
18
4 2
4
2
6
264
6
4
4
x
x
x
Completing the Square:
19.
19
Completing the Square:
Example 2
x2
- 16x +15 = 0
Re-write the equation in
the form ax + by = c
Divide by leading coefficient
x2
- 16x = -15
2
2
bTake one-half of b, then
square it. Add the square
to both sides.
x2
- 16x + 64 = -15 + 64
Simplify both sides.x2
- 16x + 64 = -15 + 64
(x - 8)2
= 49
20.
1) Find the value of
2
2
b
2
x bx
Completing the Square:
2. Add the value to the
expression, this completes
the square
2
6x x
2
10x x
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