P.o.D. – Solve each system
3𝑧 − 6𝑤 = 15
1.) {
0.5𝑧 − 𝑤 = 22
2𝑥 + 10𝑦 = 16
2.) {
𝑥 = −3𝑦
𝑎 = 2𝑏 − 4
3.) { 𝑏 = 2𝑐 + 2
𝑐 = 4𝑎 + 6
1.) No Solution
2.) X= -12, y=4
3.) a= -8/5, b= 6/5, c= -2/5
6-1: Quadratic Functions
Learning Targets: be able to expand
products and squares of binomials.
Quadratic Expression: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Quadratic Equation: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
Quadratic Function:
𝑓 (𝑥 ) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
*All three of these are known as standard
form of a quadratic.
If we had a quadratic in two or more
variables, it would be written as
𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹. We
will use this later in the year when we
study The Conic Sections.
Review: FOIL {Binomial Expansion}
First Outer Inner Last
Binomial – an expression with two terms.
EX: FOIL (x-3)(x+8)
𝑥 2 + 8𝑥 − 3𝑥 − 24
= 𝑥 2 + 5𝑥 − 24
EX: A portrait is 20 centimeters by 90
centimeters. A frame around the portrait
is f centimeters wide. Write the total area
of the portrait and the frame in standard
form.
Begin by drawing a picture.
(show on the whiteboard)
Write a product of the length times the
width.
(20 + 2𝑓)(90 + 2𝑓)
FOIL these binomials.
1800 + 40𝑓 + 180𝑓 + 4𝑓 2
= 4𝑓 2 + 220𝑓 + 1800
*Note: it is customary to write your
answer in descending order (highest
exponent to lowest).
EX: Write the area of a square with sides
of length (2a+b) in standard form.
(draw a picture on the whiteboard)
𝑙 (𝑤) = 𝑠𝑖𝑑𝑒 (𝑠𝑖𝑑𝑒)
= (2𝑎 + 𝑏)(2𝑎 + 𝑏)
= 4𝑎2 + 2𝑎𝑏 + 2𝑎𝑏 + 𝑏 2
= 4𝑎2 + 4𝑎𝑏 + 𝑏 2
Binomial Square Theorem:
(𝑥 + 𝑦)2 = 𝑥 2 + 2𝑥𝑦 + 𝑦 2
(𝑥 − 𝑦)2 = 𝑥 2 − 2𝑥𝑦 + 𝑦 2
Use the Binomial Square Theorem to
find the following:
a.) (3𝑥 + 𝑦)2
b.) (2𝑥 − 4𝑦)2
a.) (3𝑥)2 + 2(3𝑥 )(𝑦) + (𝑦)2 =
9𝑥 2 + 6𝑥𝑦 + 𝑦 2
b.) (2𝑥)2 − 2(2𝑥 )(4𝑦) + (−4𝑦)2 =
4𝑥 2 − 16𝑥𝑦 + 16𝑦 2
http://www.youtube.com/watch?v=w8smA_akWBY
http://www.youtube.com/watch?v=Axv7cqezipY
EX: A large circular pipe coming up
from the ground is surrounded by a
circular region of drainage stones. The
distance from the edge of the pipe to the
outer edge of the drainage stones is w
feet, and the radius of the drainage
stones, including the large pipe, is 7 feet.
a. Write a quadratic expression in
standard form for the area of the
opening of the circular pipe, not
including the drainage stones.
b. How many square feet are covered
by drainage stones, in terms of w?
a. Draw a picture (draw on the
whiteboard)
Next, find the radius of the inner pipe.
r = (7-w). Apply this to the formula for
the area of a circle, 𝐴 = 𝜋𝑟 2 .
𝐴 = 𝜋(7 − 𝑤)2
Expand this polynomial.
𝐴 = 𝜋(7 − 𝑤)(7 − 𝑤)
= 𝜋(49 − 7𝑤 − 7𝑤 + 𝑤 2 )
= 𝜋(49 − 14𝑤 + 𝑤 2 )
= 49𝜋 − 14𝜋𝑤 + 𝜋𝑤 2
= 𝜋𝑤 2 − 14𝜋𝑤 + 49𝜋
b. First find the area of the pipe and the
drainage stones.
𝐴 = 𝜋𝑟 2 = 𝜋(7)2 = 49𝜋
Now, subtract the area of the pipe (which
we previously calculated).
𝐴 = 49𝜋 − (𝜋𝑤 2 − 14𝜋𝑤 + 49𝜋)
= −𝜋𝑤 2 + 14𝜋𝑤
= 14𝜋𝑤 − 𝜋𝑤 2
EX: A city park wants to build a brick
walkway around a rectangular flower
garden. The garden is 6ft wide by 25ft
long. Find an expression for the area of
the walkway if it is x feet wide.
(Draw a picture)
First, find the area of the walkway and
the flower garden combined.
(2𝑥 + 6)(2𝑥 + 25)
= 4𝑥 2 + 50𝑥 + 12𝑥 + 150
= 4𝑥 2 + 62𝑥 + 150
Now, find the area of the flower garden.
6(25) = 150
The area of the walkway will be the
difference of the previous two areas.
𝐴 = 4𝑥 2 + 62𝑥 + 150 − 150
= 4𝑥 2 + 62𝑥
Suppose each brick covers ¼ square foot.
How many more bricks are needed to
make a 5ft walkway than to make a 4ft
walkway?
Begin by finding the area of a 5ft
walkway.
𝐴(5) = 4(5)2 + 62(5)
= 410 𝑠𝑞𝑢𝑎𝑟𝑒 𝑓𝑒𝑒𝑡
Now find the number of bricks needed.
410
= 1640
1⁄
4
Next, find the area of a 4ft walkway.
𝐴(4) = 4(4)2 + 62(4) = 312
Now find the number of bricks needed.
312(4) = 1248
Finally, find the difference between the
number of bricks needed for a 5ft
walkway and the number of bricks
needed for a 4ft walkway.
1640-1248=392 more bricks.
Try the following on your own:
a.) (x-3)(x+4)
b.) (3n+1)(2n-5)
c.) (3x-y)(3x+y)
d.) (𝑥 + 5)2
a.) 𝑥 2 + 4𝑥 − 3𝑥 − 12 = 𝑥 2 + 𝑥 − 12
b.) 6𝑛2 − 15𝑛 + 2𝑛 − 5 =
6𝑛2 − 13𝑛 − 5
c.) 9𝑥 2 + 3𝑥𝑦 − 3𝑥𝑦 − 𝑦 2 = 9𝑥 2 − 𝑦 2
d.) 𝑥 2 + 2(𝑥 )(5) + 52 =
𝑥 2 + 10𝑥 + 25
Upon completion of this lesson, you
should be able to:
1. Identify the different forms of a
quadratic function.
2. Expand binomials (FOIL).
3. Apply binomial expansion to story
problems.
For more information, visit
https://www.youtube.com/watch?v=qgtUXG4r_wM
HW Pg.377 2-28E