Alg1 Lesson 3.9: Interpreting the Discriminant

Lesson 3.9: Interpreting the

Discriminant

Obj: 7B, 8A, 11A: I can use the discriminant to
determine how the roots will appear.

EQ: How do I know if a quadratic equation has 2
unique real, 1 repeated real, or 2 imaginary
solutions?

Roles:
Facilitator
Scribe
Resourcer
Includer

Lesson Goals:
● Creative Thinking
● Talk through controversies and conflict
● Recognize and reduce ambiguity
● Encourage thinking based on formulas and prior info
● Help explain ideas to each other
● Own your ideas and work
● Record ideas in your journal
● Answer Questions on Slides
● Follow your team roles

Facilitator

• Make sure that all peers are staying on task.

• Give advice or suggestions to resolve the problem.

• Be sure everyone is able to explain.

Scribe

• Make sure peers organize their results on their own papers.

• Remind peers to use color, arrows, and other math tools to
communicate your mathematics, reasons, and connections.

• Be ready to join the teacher for a huddle.

Resourcer

• Make sure peers are getting the materials needed.

• Make sure that all materials are put away neatly.

• Make sure that peers are logged in to the needed site.

• Help troubleshoot any technology difficulties that may arise.

Includer

• Make sure that all peers are talking about their work.

• Helps keep peers’ voice volume low.

• Encourages everyone to ask questions.

• Communicates conflicts or questions to the teacher.

Poll

Which role are you doing today?

• Scribe

• Includer

• Facilitator

• Resourcer

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

Remember to work on the following too…

Part 1: Parabolas Without

x-intercepts

What are the x-intercepts of the graph of the
function f (x)= x² +1?

Does the equation x² + 1 = 0
have any real-number
solutions? Why or why not?

Multiple Choice

What happens when you square a positive number? A negative number? Zero?

• The product
  will always be nonnegative.

• Positive, negative, zero.

• The sign varies.

• Always a negative.

Multiple Choice

What happens if you add 1 to a nonnegative number?

• The sum is always greater than 1.

• The sum can be negative.
• The sum is always less than 1.
• The sum is always greater than or equal to 1.

Multiple Choice

If you square a real number x and add 1 to it, what can you say about the resulting value?

• The resulting value is always equal to 1.
• The resulting value can be negative.
• The resulting value is always greater than or equal to 1.

Drag and Drop

What kinds of values will be the outputs of f (x)= x^2 + 1? The outputs will always be ​

Placeholder

. Will there be a real value for x that will make x^2 +1= 0? ​

Placeholder

, since x^2 +1 >= 1 and
therefore ​

Placeholder

equal to 0.
What does this mean about the x-int of f(x) = x^2 + 1? There ​

Placeholder

x-int

Drag these tiles and drop them in the correct blank above

1 or greater

1 or less

0 or greater

0 or less

No

Yes

is never

is sometimes

are no

are 2

Use these when reasoning frequently!!

Solving f (x)= x² +1 using the Quadratic Formula

√-4 is NOT a real number!

Drag and Drop

What does it mean that √9 = 3? The square root of ​

Placeholder

is ​

Placeholder

, which means that ​

Placeholder

^​

Placeholder

is equal to ​

Placeholder

.

Drag these tiles and drop them in the correct blank above

9

3

2

Drag and Drop

If √−4 = z, then what is another way of writing that, without using the
square root symbol? Another way to write it is ​

Placeholder

^​

Placeholder

= ​

Placeholder

.
(2)^2 = ​

Placeholder

and (−2)^2 =
​

Placeholder

.

Drag these tiles and drop them in the correct blank above

z

2

−4

4

8

Multiple Choice

What real number can you square to give you −4?

• It is not
  possible to find a real number whose square is −4.

• 2

• -2

• -2j

Discriminant: The Value Inside the Square Root

Part 2: How Many

Solutions?

2 Imaginary Solutions

(No Real Solution)

b² - 4ac < 0

1 Repeated Real

Solution

b² - 4ac = 0

2 Unique Real

Solutions

b² - 4ac > 0

Discriminant = b² - 4ac

Handout 3.9: Practice Using the Discriminant

Set up each quadratic equation into the Quadratic
Formula. Identify the value of the discriminant and
state whether the equation has two unique real, one
repeated real, or two non-real solutions.

Part 3: Determining How

Many Intersections

Is it possible
to determine
the
intersection
points
algebraically,
and if so,
how?

Use substitution to set each quadratic expression equal to each other.

The x values of the intersections are x = 0.485 and x = 2.943

The discriminant > 0 and there are 2 intersection points.

What do you think would happen if the discriminant < 0 and if the discriminant = 0?

The discriminant
for both equations
together is negative
and there are no
intersections.

If the discriminant was
0, there would only be
1 intersection.

Random Question of the Day Time

https://wheelofnames.com/4ke-epz We’ll spin
the wheel as a class and spend a minute or so
discussing our answers.