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Algebra II
Quadratic Expressions
and Equations
Quadratic Expressions
There are several examples of quadratic
equations on p.345. Generally, “quadratic”
means the degree is two. The degree of an
expression is the number of times a variable
appears as a factor.
The final equation, A = xy, has two independent
variables. The others contain one independent
variable.
Quadratic Expressions
There are several definitions on p.346.
ax2 + bx + c is a quadratic expression in one
variable. This is called the standard form of a
quadratic.
ax2 + bx + c = 0 is a quadratic equation in one
variable.
f(x) = ax2 + bx + c is a quadratic function in one
variable.
Ax2 + Bxy + Cy2 + Dx + Ey + F is supplied as an
example of a quadratic expression in two
variables. We’ll see a few of these, but most
examples will contain one independent variable.
Quadratic Expressions
Fill the blank.
9
so
32 = _____
= _____
3
9 has another square root. What is it? -3
(-4)2 = _____
16
so
= _____
4
Every positive number has two real square
roots.
means the positive square root. It’s
called the “principal square root”.
-4 and 4 are both square roots of 16. 4 is the
principal square root.
Quadratic Expressions
(2x)2 = _____
4x2
so
= _____
2x
I thought you’d say that. We’ll return to this
example in a moment.
Here’s why. Replace x in
with -4.
That’s not the -4 we put in for x.
This insures the principal square root is positive.
Now back to
2|x|
= _____
Quadratic Expressions
Remember the FOIL rule?
First – Outside – Inside – Last
It is the way we multiply binomials.
Multiply.
(x + 5)(x + 6)
x2 + 11x + 30
(2x + 5y)(3x – y)
(5 – n)(3 – 2n)
6x2 + 13xy - 5y2
15 - 13n + 2n2
Quadratic Expressions
(x + 7)(x + 7) can be written (x + 7)2. You can
apply the FOIL rule, but our book lists another
property, the Binomial Square Theorem.
(x + y)2 = x2 + 2xy + y2
(x – y)2 = x2 – 2xy + y2
Apply the Binomial Square Theorem.
(x + 5)2
(2x – 3)2
x2 + 10x + 25
4x2 - 12x + 9
Quadratic Expressions
A walkway w feet wide will be built around a
rectangular garden 30’ x 15’. Draw a sketch.
15 + 2w
Write an expression
for the length and
width of the outside
rectangle.
30 + 2w
Write an expression in factored form for the area
of the entire figure. (15 + 2w)(30 + 2w)
Write a quadratic expression in standard form
for the area of the entire figure. 450 + 90w + 4w2
Quadratic Expressions
A bulletin board measures 8’ x 3’. A border of
uniform width will be placed around the edge of
the board. Write a quadratic expression in
standard form for the available region remaining
on the bulletin board.
(8 – 2w)(3 – 2w)
= 24 – 22w + 4w2
Quadratic Expressions
Write the area of a square with sides of length
a + b in standard form. Draw a picture of the
square.
a2 + 2ab + b2
Quadratic Equations
Our book supplies this piecewise definition of
absolute value.
I like to say the absolute value of a number is
the distance between 0 and that number on
the number line. Absolute value is always nonnegative.
Quadratic Equations
Solve for x: |x – 2| = 5.3
The highlighted part could either equal 5.3 or
-5.3 to make this equation true.
x – 2 = 5.3 or x – 2 = -5.3
x = 7.3 or -3.3
The problem can also be thought of this way…
You’re asked to find the numbers on the
number line that are 5.3 units from 2.
Again, 7.3 or -3.3.
Quadratic Equations
What will the graph of
look like? Picture
the graph, then graph on your calculator.
Is this what you expected?
Why does it look like this?
Remember
You just graphed y = |x|.
Here’s the first definition we had for |x|.
Quadratic Equations
Solve x2 = 12 for x.
Approximate these solutions
to the nearest hundredth.
x ≈ + 3.46 Check
You wrote these values in “simple radical form”
in Geometry class. We’ll review this briefly.
More later.
Find the largest square factor of 12.
Quadratic Equations
Solve (n – 3)2 = 18 for n. Write these answers
both as decimals,
and in simple radical
form. They are not
opposites.
n ≈ 7.24 or -1.24 Check.
Quadratic Equations
Solve |3x + 1| = 5 for x.
3x + 1 = 5 or 3x + 1 = -5
x = 4/3 or x = -2
For what real numbers does:
|x| = |-x| ?
All real numbers.
|x| = -|x| ?
0
Quadratic Equations
For problems 16 – 21 on the homework, you
need to know the definition of rational and
irrational numbers.
The root word of rational is ratio. A ratio is a
fraction. Rational numbers are ones that can be
written as a typical fraction, an integer divided
by a non-zero integer.
An irrational number is one that cannot be
written as such. π is probably the most famous
irrational number. The decimal, like all irrational
numbers, never ends and never has a block of
digits that repeats.
Quadratic Equations
Regarding radicals, the number is rational if we
take the square root of a “square number”:
1, 4, 9, 16, 25, 36, … because the result will be
an integer, which can be expressed as a fraction
over 1.
are examples we’ve seen
that are irrational.
Quadratic Equations
Simplify
=5
Graph y = |x – 3| with a table of values.
I often use these x-values.
Remember y = |x|? It was v-shaped.
Try these.
Quadratic Equations
A rectangle is 6 in x 12 in. What is the diameter
of a circle with the same area?
The rectangle has
area 72 in2. Replacing
A, and solving for r…
The diameter of a
circle with the same
area as a 6” x 12”
rectangle is about
9.57 in.
≈ 4.79