Download The Discriminant and Complex Numbers

Describing the Roots of a Quadratic Equation
You will investigate the relationship between the roots of a quadratic equation and the discriminant of the
quadratic equation. The discriminant is the expression b2 –4ac that comes from the Quadratic Formula
 b  b 2  4ac
x=
2a
Discriminant D = b2 – 4ac
As you will see, the discriminant gives useful information about the roots of the quadratic function.
Step 1 – Use the Quadratic Formula to find the roots of each quadratic function below.
1.) y = 2x2 - 9x + 4
2.) y = x2 + 2x + 1
3.) y = 2x2 -2x + 1
Step 2 – Calculate the discriminant (b2 – 4ac) of each of the above functions .
1.)
2.)
3.)
Step 3 – Graph each of these equations using the TI-83 Graphics Calculator. Can you predict the number of
x-intercepts of the quadratic function when the discriminant is known? Explain.
Nature of the Roots
The discriminant allows us to describe the nature of the roots of a quadratic function without actually finding
the roots. When one describes the nature of the roots, they are describing two things:
i) how many roots the function has (1 or 2)
ii) the kind of roots that they are (real or imaginary)
Three Cases:
If D>0, then you’ll have two real roots (two x-intercepts)
If D=0, then you’ll have one real root (one x-intercept because the vertex is
on the x-axis)
3) If D<0, then you’ll have two imaginary roots (no x-intercepts – because the parabola does not touch
the x-axis)
1)
2)
Try these examples:
Determine the nature of the roots of a)
y   x 2  5x  6 .
b)
y  2x2  4x  4 .
Other Questions Involving the Discriminant.
kx 2  x  4  0 have two real roots?
2
For what value of k will the function 4 x  4 x  k  0 have imaginary roots?
2
For what value of m will the function 4 x  5 x  m  1  0 have 1 real root?
a) For what value of k will the function
b)
c)
Complex Numbers
A need for Complex Numbers arose when mathematicians could not take the square roots of a negative number.
They developed the Complex Number System to formally describe imaginary numbers. We will need the
Complex Number System to describe the roots of quadratic functions that do not intersect the x-axis.
Example: Find the roots of y
x
 2 x 2  5x  6.
 b  b 2  4ac  (5)  (5) 2  4( 2)(6)

2a
4
 5  25  48

4
 5   23

4
We know the roots are Complex, but we cannot simplify the
 23 in the Real Number System.
Mathematicians introduced a new symbol variable, i, an imaginary number, to represent the  1 .
In short, i   1 in the Complex Number System
Therefore, the above solution would simplify to:
x
 5   1 23  5  i 23

4
4
Try these:
Represent each of the following in the simplest form using the Complex Number System
a)
2 9
4
b)
 4   32
4
c)
3 6
5
d) The roots of y = -6x2 + 4x -3
Solution,