Unit 4: Quadratic Functions
Name ____________________________
Lesson 3:
Exploring Zeros, Quadratic Equations, and Quadratic Functions
Day/Period___
Square Roots and Imaginary Numbers
When you have a negative under a square root, you factor out the  1 and then simplify the
 1 as i.
i  1
We call these types of square roots imaginary numbers. By using the letter i we can work with our
answers, knowing they don’t really “exist”.
Simplify the following:
250
2)
0
3)  4
5)  48
6)
 48
7)
1)
9)
250m 6 n 5
10)
 250
25m 2 n 2
11) Extension:
4)
4
8)
28m 6 n 4
3
125m3 n 6
TYPES of ZEROS
When we graph quadratic functions we get a parabola. Depending on where the graph is located, we
may see two, one, or NO zeros at all.
For each of the following, find the zeros using the quadratic formula. Give both exact and decimal
solutions if the answers are irrational.
1) y  6 x 2  9 x  6
2) y  3x 2  12 x  12
What is the value of the discriminant? _____
What is value of the discriminant? _____
Zeros are _______________________
Zeros are ______________________.
3) y  3x 2
4) y   x 2  5 x  9
What is the value of the discriminant? _____
What is the value of the discriminant? _____
Zeros are _______________________
Zeros are _______________________
5) y  x 2  4 x  8
6) y  3x 2  12 x  16
What is the value of the discriminant? _____
What is the value of the discriminant? _____
Zeros are _______________________
Zeros are _______________________
Summary: The Discriminant
If the value b 2  4ac  0 there will be _____ real solutions.
If b 2  4ac is a perfect square, the two solutions will be _____________________
If b 2  4ac is NOT a perfect square, the two solutions will be _____________________
If the value b 2  4ac  0 there will be _____ real solutions but both solutions will be ______________.
If the value b 2  4ac  0 there will be _____ IMAGINARY solutions.
Note: If the solutions are RATIONAL, then the quadratic equation could have also been factored
to solve it!
Question: For which of the problem(s) from #1-6 could we have ALSO used the zero-product property
to find the zeros?
Use factoring to find the zeros again from problems # ___________________:
Extension Vocabulary: “Roots” are answers to equations that are set equal to zero.
Identify a, b, and c and find the Discriminant.
Solve using the Quadratic formula.
Simplify all radicals completely and find both exact and decimal answers.
1)
0  2 x 2  3x  5
2) 3x 2  5 x  1  0
3)
x 2  6x  4  0
4) 0   x 2  2 x  7
5)
0  2x 2  4x  6
6) 6 x 2  x  2  0
  5  13 


6


1  i 2
2 1
 , 
3 2
 1  2 2
5

 1, 
2

3  5
Which problems above could have been solved using factoring and the zero-product property instead?