Aim #66: What are the nature of the roots (discriminant)?
Homework: Worksheet
Do Now: Solve the following using the quadratic formula.
16x2 - 8x + 1 = 0
Recall the quadratic formula:
2
The discriminant of the quadratic formula is b - 4ac
It tells us how many roots the quadratic will have and the nature
of those roots!
If:
2
b - 4ac > 0 and perfect
square
2
b - 4ac > 0 and not a
perfect square
2
b - 4ac = 0
2
b - 4ac < 0
then the roots are:
then the graph looks
like:
Some notes about the discriminant and nature of the roots:
- When the discriminant is a perfect square (0 included) this tells us that the
given quadratic is factorable!
- When the discriminant is 0, we only have one solution. We sometimes refer to
this as a double root.
Determine both the nature of the roots and the properties of the discriminant
from the graphs given below.
Describe the roots of the following equations and how many timesthe graph
would cross the x - axis. Determine if we can factor it or have to use one of the
other methods to solve for the roots. (quadratic formula/completing the square)
2
1. 3x + 5x - 2 = 0
2
3. x - 5x = 5
2
2. 4x = 4x - 1
2
4. x - 10x = - 40
For each of the following, find the value of the discriminant. Then choose one of
the following:
A) The equation has 2 real rational roots
B) The equation has 2 real irrational roots
C) The equation has 2 real equal roots (one double root)
D) The equation has 2 imaginary roots
2
2) 2x + 7x + 6 = 0
2
4) x + 10 = 0
1) x + 5x - 8 = 0
3) x - 11 = 0
2
5) -2x +2x - 6 = 0
2
2
2
6) 4x + 12x + 9 = 0
For each of the following, find the value of the discriminant. Then choose
one of the following:
A) The parabola has 2 x-intercepts
B) The parabola has 1 x-intercept
C) The parabola has no x-intercepts and is completely above the x-axis
D) The parabola has no x-intercepts and is completely below the x-axis
2
2) y = 2x + x - 10
2
2
4) y = x + 10
1) y = x + 5x - 8
2
3) y = x - 9
2
2
5) y = -2x - 6
6) y = x + 6x + 9
Sum It Up!
2
The discriminant of a quadratic is b - 4ac.
It can be used to determine how many solutions the quadratic has, as well as the
nature of those solutions.
We can also use this to help us determine which method of solving to use.