1. Chapter 3 Quadratic Equations
3.5
The
Discriminant

2. The Nature of the Roots
The quadratic formula
will give the roots of the quadratic equation.
From the quadratic formula, the radicand,
b2 - 4ac, will determine the Nature of the Roots.
By the nature of the roots, we mean:
whether the equation has real roots or not
if there are real roots, whether they
are different or equal
The expression b2 - 4ac is called the discriminant
of the equation ax2 + bx + c = 0 because it discriminates
among the three cases that can occur.

3. x2 + x - 6 = 0 The Nature of the Roots [cont’d] The value of the
discriminant is
greater than zero.
x = -3 or 2
There are two distinct
real roots.

4. The Nature of the Roots [cont’d]
x2 + 2x + 1 = 0
The value of the
discriminant is 0.
There are two
equal real roots.
x = -1 or - 1

5. The Nature of the Roots [cont’d]
2x2 - 3x + 8 = 0
The value of the
discriminant is
less than zero.
Imaginary roots

6. The Nature of the Roots [cont’d]
We can make the following conclusions:
If b2 - 4ac > 0, then there are two different real roots.
If b2 - 4ac = 0, then there are two equal real roots.
If b2 - 4ac < 0, then there are no real roots.
Determine the nature of the roots:
x2 - 2x + 3 = 0
4x2 - 12x + 9 = 0
b2 - 4ac = (-2)2 - 4(1)(3)
= -8
b2 - 4ac = (-12)2 - 4(4)(9)
= 0
There are two equal
real roots.
The equation has no
real roots.

7. Determine the value of k for which the equation x2 + kx + 4 = 0 has
a) equal roots b) two distinct real roots c) no real roots
a) For equal roots, b2 - 4ac = 0.
Therefore, k2 - 4(1)(4) = 0
k = 0
k2 = 16
k = + 4
The equation has equal roots
when k = 4 and k = -4.
b) For two different real roots, b2 - 4ac > 0.
k > 0
k2 > 16
Therefore, k > 4 or k< -4. This may be written as | k | > 4.
Which numbers when squared
produce a result that is greater
than 16?
Which numbers when squared
produce a result that is less
than 16?
c) For no real roots, b2 - 4ac < 0.
k2 < 16
Therefore, -4 < k < 4. This may be written as | k | < 4.