# Discriminant of a Quadratic Equation

Generally, a discriminant is a function of the coefficients of a polynomial. It determines the nature of solutions an equation can have without exactly finding them.

In a quadratic formula, the discriminant is only a part of the quadratic formula within the square root. For a quadratic equation ax2 + bx + c = 0, b2 – 4ac is the discriminant (D), as shown in the diagram below.

Thus, to find the discriminant of a quadratic equation, follow the following steps:

Step 1: Compare the given quadratic equation with its standard form ax2 + bx + c = 0 and find the values of a, b and c

Step 2: Substitute the values in the discriminant b2 – 4ac to get the result

Let us find the discriminant of the quadratic equation x2 + 10x + 16 = 0

Comparing the given quadratic equation with its standard form ax2 + bx + c = 0, we get a = 1, b = 10, c = 16

Substituting the values in the discriminant b2 – 4ac, we get

= (10)2 – 4 Ã— 1 Ã— 16

= 100 â€“ 64

= 36

Thus, the discriminant of the quadratic equation x2 + 10x + 16 = 0 is 36

What is the discriminant of the quadratic equation 3 – 4x – 6x2 = 0?

Solution:

Rearranging the equation to -6x2 – 4x + 3 = 0
Comparing the given quadratic equation with its standard form ax2 + bx + c = 0 we get, a = -6, b = -4, c = 3
Substituting the values in the discriminant b2 – 4ac, we get
= (-4)2 – 4 Ã— -6 Ã— 3
Â = -56
Thus, the discriminant of the quadratic equation 3 – 4x – 6x2 = 0 is 36

Determine the discriminant for the quadratic equation -3x2 + 4x + 1 = 0

Solution:

Comparing the given quadratic equation with its standard form ax2 + bx + c = 0 we get, a = -3, b = 4, c = 1
Substituting the values in the discriminant b2 – 4ac, we get
= (4)2 – 4 Ã— -3 Ã— 1
Â = 28
Thus, the discriminant of the quadratic equation -3x2 + 4x + 1 = 0is 36

Discriminant also tells the number of solutions a quadratic equation has and the nature of the graph it produces in the coordinate plane. There can be three different situations:

## If the Discriminant is Positive

• If b2 – 4ac > 0, the quadratic equation has 2 real solutions

It happens because the square of a positive number always gives a real number. So, when the discriminant is greater than 0 it is 2 roots that real and distinct. The graph of such an equation crosses through the x-axis at two points.

## If the Discriminant is Negative

• If b2 – 4ac < 0, the quadratic equation has no real solutions but two different complex or imaginary roots

It happens so because the square of a negative number gives an imaginary number. For example, ${\sqrt{-9}}$ = 3i. So, when the discriminant is less than 0, it has two distinct and complex roots. The graph of such an equation does not cross the x-axis.

## If the Discriminant is Zero

• If b2 – 4ac = 0, the quadratic equation has one real root

It happens so because the square of 0 is 0. The graph of such an equation touches the x-axis at one point.

Determine whether each of the following quadratic equations has two real roots, one real root, or no real roots.
(a) 2x2Â + 4x âˆ’ 6 = 0
(b) 3x2Â – 6x + 8 =0
(c) x2Â + 2x + 1 =0

Solution:

(a) Comparing the given quadratic equation with its standard form ax2 + bx + c = 0, we get a = 2, b = 4, c = -6
Substituting the values in the discriminant b2 – 4ac, we get
= (4)2 – 4 Ã— 2 Ã— -6
= 64
Since the value of discriminant is positive (> 0), hence the equation 2x2Â + 4x âˆ’ 6 = 0 has two real roots
(b) Comparing the given quadratic equation with its standard form ax2 + bx + c = 0, we get a = 3, b = -6, c = 8
Substituting the values in the discriminant b2 – 4ac, we get
= (-6)2 – 4 Ã— 3 Ã— 8
= -60
Since the value of the discriminant is negative (< 0), the equation 3x2 – 6x + 8 =0 has no real solutions but two different complex roots.
(c) Comparing the given quadratic equation with its standard form ax2 + bx + c = 0, we get a = 1, b = 2, c = 1
Substituting the values in the discriminant b2 – 4ac, we get
= (2)2 – 4 Ã— 1 Ã— 1
= 0
Since the value of the discriminant is zero (= 0), the equation x2 + 2x + 1 =0 has one real root.