Properties of Quadratics
Determining the “Nature of the Roots”
We now need to determine
the nature of the roots. In
other words, describe what
kind of roots we have. To do
that, we need to use part of
the quadratic formula:
b  24a ac
x
We know that the roots are
found where the quadratic
equation intersects the x-axis.
2 b  b2  4ac
We are going to use the part of the
quadratic formula under the radical.
This is known as the DISCRIMINANT.
Using the discriminant: b  4ac
2
There are three things we need to check for:
1. Real vs. Imaginary
2. Equal vs. Unequal
3. Rational vs. Irrational
Lets check the four cases:
Case 1: Discriminant is negative:
b 2  4ac  0
This means that the roots are
imaginary because we have a
negative under the radical.
Case 2: Discriminant is zero:
b 2  4ac  0
This means that the roots are
real, equal and rational.
EX: Suppose the formula gave
us the following:
3 0
x
6
3 0 3 1
30 3 1
x
 
x
 
6
6 2
6
6 2
So we see that both roots are ½, so they are
equal, and ½ is a real, rational number!
Case 3: Discriminant is positive
and a perfect square:
b 2  4ac  0
Case 4: Discriminant is positive
and not a perfect square:
b 2  4ac  0
This means that the roots are
real, unequal and rational.
This means that the roots are
real, unequal and irrational.
EX: Suppose the formula gave
us the following:
EX: Suppose the formula gave
us the following:
1 9
x
2
1 3 4
x
 2
2
2
1 17
x
2
1 17
x
2
1 3  2
x

 1
2
2
1 17
x
2
So we see that both roots real,
rational numbers that are NOT
equal
So we see that both roots real,
irrational numbers that are NOT
equal
Summary:
1. b  4ac  0
Discriminant is negative so roots are
imaginary. ONLY TIME THAT THE
ROOTS ARE IMAGINARY!
3. b 2  4ac  0
Discriminant is a positive perfect square
so roots are real, rational and unequal.
2. b 2  4ac  0
Discriminant is zero so roots are real,
rational and equal. ONLY TIME THAT
THE ROOTS ARE EQUAL!
4. b 2  4ac  0
Discriminant is a positive non-perfect
square so roots are real, irrational and
unequal. ONLY TIME THE ROOTS
ARE IRRATIONAL!
2
Page 1
#1:
x 2  5x  4  0
a 1
b  5
c  4
b 2  4ac
 52  41 4
41
The discriminant is a
positive, non-perfect square.
Therefore the roots are:
1. Real
2. Irrational
3. Unequal
#2:
x 2  8 x  20  0
a 1
b 2  4ac
b  8
c  20  82  41 20
144
The discriminant is a
positive, perfect square.
Therefore the roots are:
1. Real
2. Rational
3. Unequal
Page 1
#5:
4 x 2  12x  9  0
a4
b  12
c9
b 2  4ac
122  449
0
The discriminant is zero.
Therefore the roots are:
1. Real
2. Rational
3. Equal
2x2  6x  5
 6x  5  6x  5
#7:
2x2  6x  5  0
a2
b  6
c5
b 2  4ac
 62  425
4
The discriminant is negative.
Therefore the roots are:
1. Imaginary
Page 1
#8:
3x 2  5 x  0
a3
b  5
c0
b 2  4ac
 52  430
25
The discriminant is a
positive, perfect square.
Therefore the roots are:
1. Real
2. Rational
3. Unequal
x2  4x  3
 4x  3  4x  3
#22:
x2  4x  3  0
a 1
b  4
c  3
b 2  4ac
 42  41 3
28
The discriminant is a
positive, non-perfect square.
Therefore the roots are:
1. Real
2. Irrational
3. Unequal
Homework
•Page 1
#3,6,9,11,13,15,17,19,
21,24,27,30,33