2. We have already studied the quadratic formula,
x =
−b ± b
2
– 4ac
2a
The binomial inside the radical sign is
called the discriminant. It is used to determine
the nature of the roots of a quadratic equation.
We can also determine the number of real roots
for a quadratic equation with this number. The
following table will give us the relation between
the discriminant and the nature of the roots.
3. Discriminant
Nature of the
Roots
Number of
real roots
b2 – 4ac = 0
Real and
Equal
1
b2 – 4ac > 0 and a
perfect square
Rational and
Unequal
2
b2 – 4ac > 0 but not a
perfect square
Irrational and
Unequal
2
b2- 4ac < 0
Imaginary/No
Real Roots
None
4. We will discuss here about the different cases
of discriminant to understand the nature of the roots
of a quadratic equation.
We know that x1 and x2 are the roots of the
general form of the quadratic equation ax2 + bx +
c = 0 where (a ≠ 0) then we get
x1=
−b + b2 – 4ac
2a
and x2=
−b – b2 – 4ac
2a
5. Here a, b and c are real and rational.
Then, the nature of the roots x1 and x2 of
equation 𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0 depends on the
quantity or expression i.e., b2 – 4ac under the
radical sign.
Thus, the expression b2 – 4ac is called the
discriminant of the quadratic equation 𝑎𝑥2
+
𝑏𝑥 + 𝑐 = 0.
6. Generally we denote discriminant of
the quadratic equation by ‘∆‘ or ‘D’. Therefore,
Discriminant ∆ = b2 − 4ac.
Depending on the discriminant we shall
discuss the following cases about the nature of
roots x1 and x2 of the quadratic equation 𝑎𝑥2
+
𝑏𝑥 + 𝑐 = 0.
When a, b and c are real numbers, a ≠
7. Case I: b2 – 4ac = 0
When a, b and c are real numbers, a ≠ 0 and
discriminant is zero (i.e., b2 − 4ac = 0), then the roots
x1 and x2 of the quadratic equation 𝑎𝑥2
+ 𝑏𝑥 + 𝑐 =
0 are real and equal.
Example 1:
Find the discriminant value of x2 –12x + 36 = 0 and
determine the number of real roots.
Since the discriminant value of the equation is zero
then the equation x2 – 12x + 36 = 0 has a double root
and the roots are real and are equal.
9. Example 2: Find the nature of the roots of the
equation x² – 18x + 81 = 0.
Clearly, the discriminant of the given quadratic equation
is zero and coefficient of x2 and x are rational.
Answer: Therefore, the roots of the given quadratic
equation are real and equal.
10. Case II: b2 - 4ac > 0 and perfect square
When a, b and c are real numbers, a ≠ 0 and
discriminant is positive and perfect square, then the
roots x1 and x2 of the quadratic equation ax2 + bx + c =
0 are real, rational, unequal.
Example 3: Find the discriminant value for the
equation x2 + 5x + 6 = 0 and determine the number of
real roots.
11. Describe the nature of the roots:
Since the discriminant value of the equation is
greater than 0 and a perfect square, then there are two
real roots of the equation x² + 5x + 6 = 0 and the roots
are rational numbers but not equal.
Answer: The roots of the quadratic equation x² +5x +
6 = 0 are -3 and -2.
12. Example 4:
Find the nature of the roots of the equation
3x2 – 10x + 3 = 0 without actually solving them.
Clearly, the discriminant of the given quadratic
equation is positive and a perfect square.
Therefore, the roots of the given quadratic equation
are rational and unequal.
13. Case III: b2 – 4ac > 0 and not perfect square
When a, b and c are real numbers, a ≠ 0 and
discriminant is positive (i.e., b2 – 4ac>0) but not a
perfect square then the roots of the quadratic equation
ax2 + bx + c = 0 are real, irrational and unequal. Here
the roots x1 and x2 form a pair of irrational conjugates.
Example 5: Describe the nature of the roots of the
quadratic equation 2x2 – 8x + 3 = 0.
14. Clearly, the discriminant of the given quadratic
equation is positive but not a perfect square.
Answer: Therefore, the roots of the given quadratic
equation are irrational and unequal.
15. Case IV: b2 - 4ac < 0
When a, b and c are real numbers, a ≠ 0 and
discriminant is negative (b2 - 4ac < 0), then the roots
x1 and x2 of the quadratic equation ax2 + bx + c =
0 are unequal and imaginary. Here the roots x1 and x2
are a pair of the complex conjugates.
Example 6:
Find the discriminant value 2x2 + x + 3 = 0 and
determine the number of real roots.
16. Since the discriminant value of the equation is less
than zero then the equation 2x2 + x + 3 = 0 has no
real roots or imaginary. Also, the graph of this
equation does not touch the x-axis.
