Discriminant
How to find the nature of the roots of a quadratic equation by using its discriminant: formula, 4 examples, and their solutions.
Formula
Formula
For a quadratic equation
ax2 + bx + c = 0
(a ≠ 0),
x = [-b ± √b2 - 4ac] / 2a.
Quadratic Formula
The discriminant, D, is the number
inside the radical sign:
b2 - 4ac.
Nature of the Roots
D = b2 - 4ac determines
the nature of the roots.
So, without solving the quadratic equation,
you can check the nature of the roots
by finding D.
If D is plus and is a perfect square,
then it has two rational roots.
If D is plus and is not a perfect square,
then it has two irrational roots.
If D = 0,
then it has one real root.
If D is minus (< 0),
then it has no real roots.
(= two complex roots)
Example 1
Example
Solution
The given quadratic equation is
1x2 + 7x + 10 = 0.
a = 1
b = +7
c = +10
Then D = 72 - 4⋅1⋅10.
72 = 49
-4⋅1⋅10 = -40
49 - 40 = 9 = 32
D = 32
D is plus.
And D is a perfect square.
Then the quadratic equation has
two rational roots.
So [two rational roots] is the answer.
Example 2
Example
Solution
The given quadratic equation is
1x2 - 4x - 1 = 0.
a = 1
b = -4
c = -1
Then D = (-4)2 - 4⋅1⋅(-1).
(-4)2 = 16
-4⋅1⋅(-1) = +4
16 + 4 = 20
D = 20
D is plus.
And D is not a perfect square.
Then the quadratic equation has
two irrational roots.
So [two irrational roots] is the answer.
Example 3
Example
Solution
The given quadratic equation is
1x2 - 6x + 9 = 0.
a = 1
b = -6
c = +9
Then D = (-6)2 - 4⋅1⋅9.
(-6)2 = 36
-4⋅1⋅9 = -36
36 - 36 = 0
D = 0
Then the quadratic equation has
one real root.
So [one real root] is the answer.
Example 4
Example
Solution
The given quadratic equation is
1x2 + 2x + 5 = 0.
a = 1
b = +2
c = +5
Then D = 22 - 4⋅1⋅5.
22 = 4
-4⋅1⋅5 = -20
4 - 20 = -16
D = -16
D is minus.
Then the quadratic equation has
no real roots.
So [no real roots] is the answer.