In the quadratic formula, the expression under the radical b2 - 4ac is known as the discriminant, with the condition that a ≠ 0. This quantity determines whether the roots of a quadratic equation are rational, irrational or imaginary. If the discriminant is positive, there are 2 different real number roots. If the discriminant is negative, there are 2 different imaginary roots. If the discriminant is 0, there is one repeated rational root, known as a double root. If the discriminant is a perfect square, there are 2 different rational roots and if the discriminant is positive and not a perfect square, there are two different real, irrational roots.
Examples: Determine the number and nature of roots for each of the
following quadratic equations.
1. x2 + 12x + 36 = 0
a = 1, b = 12, c = 36
4ac = 122 - 4(1)(36
= 144 - 144
equals 0, therefore there is one repeated rational root, a double root.If we
solve the equation, we'll find that the double root is x = -6.
When the discriminant is 0, the radical portion of the quadratic formula
is 0, leaving only a single value for x. Since the equation is a 2nd degree
equation, there must be 2 solutions, therefore the solution is a double root.
2. x2 - 3x + 2 = 0
a = 1, b = -3, c = 8
- 4ac = (-3)2 - 4(1)(8)
= 9 - 32
The discriminant is negative, therefore there are 2 different imaginary
roots. The roots are imaginary because the square root of a negative number is
an imaginary number.
3. 2x2 - 8x + 7/2 = 0
a = 2, b = -8,
c = 7/2
b2 - 4ac = (-8)2 - 4(2)(7/2)
64 - 28
The discriminant is positive and since 36 is a perfect
square, we have two different rational roots. The square root of 36 is 6 and -6,
which are the two different rational roots.
4. 3x2 - 7x + 1 =
a = 3, b = -7, c = 1
b2 - 4ac = (-7)2 - 4(3)(1)
= 49 - 12
The discriminant is positive and not a perfect square, therefore there are
two different real, irrational roots.
As you can see, the definition and
use of the discriminant is fairly straight forward and easy to use. This article
should have erased any confusion on this topic.