Discriminant
The discriminant is widely used in the case of quadratic equations and is used to find the nature of the roots. Though finding a discriminant for any polynomial is not so easy, there are formulas to find the discriminant of quadratic and cubic equations that make our work easier.
Let us learn more about the discriminant along with its formulas and let us also understand the relation between the discriminant and the nature of the roots.
1.  What is Discriminant in Math? 
2.  Discriminant Formula 
3.  How to Find Discriminant? 
4.  Discriminant and Nature of the Roots 
5.  FAQs on Discriminant 
What is Discriminant in Math?
Discriminant of a polynomial in math is a function of the coefficients of the polynomial. It is helpful in determining what type of solutions a polynomial equation has without actually finding them. i.e., it discriminates the solutions of the equation (as equal and unequal; real and nonreal) and hence the name "discriminant". It is usually denoted by Δ or D. The value of the discriminant can be any real number (i.e., either positive, negative, or 0).
Discriminant Formula
The discriminant (Δ or D) of any polynomial is in terms of its coefficients. Here are the discriminant formulas for a cubic equation and quadratic equation.
Let us see how to use these formulas to find the discriminant.
How to Find Discriminant?
To find the discriminant of a cubic equation or a quadratic equation, we just have to compare the given equation with its standard form and determine the coefficients first. Then we substitute the coefficients in the relevant formula to find the discriminant.
Discriminant of a Quadratic Equation
The discriminant of a quadratic equation ax^{2 }+ bx + c = 0 is in terms of its coefficients a, b, and c. i.e.,
 Δ OR D = b^{2 }− 4ac
Do you recall using b^{2 }− 4ac earlier? Yes, it is a part of the quadratic formula: x = \(\dfrac{b \pm \sqrt{b^{2}4 a c}}{2 a}\). Here, the expression that is inside the square root of the quadratic formula is called the discriminant of the quadratic equation. The quadratic formula in terms of the discriminant is: x = \(\dfrac{b \pm \sqrt{D}}{2 a}\).
Example: Find the discriminant of the quadratic equation 2x^{2}  3x + 8 = 0.
Comparing the equation with ax^{2 }+ bx + c = 0, we get a = 2, b = 3, and c = 8. So the discriminant is,
Δ OR D = b^{2 }− 4ac = (3)^{2}  4(2)(8) = 9  64 = 55.
Discriminant of Cubic Equation
The discriminant of a cubic equation ax^{3 }+ bx^{2 }+ cx + d = 0 is in terms of a, b, c, and d. i.e.,
 Δ or D = b^{2}c^{2 }− 4ac^{3 }− 4b^{3}d − 27a^{2}d^{2 }+ 18abcd
Example: Find the discriminant of the cubic equation x^{3}  3x + 2 = 0.
Comparing the equation with ax^{3 }+ bx^{2 }+ cx + d = 0, we have a = 1, b = 0, c = 3, and d = 2. So its discriminant is,
Δ or D = b^{2}c^{2 }− 4ac^{3 }− 4b^{3}d − 27a^{2}d^{2 }+ 18abcd
= (0)^{2}(3)^{2 }− 4(1)(3)^{3 }− 4(0)^{3}(2) − 27(1)^{2}(2)^{2 }+ 18(1)(0)(3)(2)
= 0 + 108  0  108 + 0
= 0
Discriminant and Nature of the Roots
The roots of a quadratic equation ax^{2 }+ bx + c = 0 are the values of x that satisfy the equation. They can be found using the quadratic formula: x = \(\dfrac{b \pm \sqrt{D}}{2 a}\). Though we cannot find the roots by just using the discriminant, we can determine the nature of the roots as follows.
If Discriminant is Positive
If D > 0, the quadratic equation has two different real roots. This is because, when D > 0, the roots are given by x = \(\dfrac{b \pm \sqrt{\text { Positive number }}}{2 a}\) and the square root of a positive number always results in a real number. So when the discriminant of a quadratic equation is greater than 0, it has two roots which are distinct and real numbers.
If Discriminant is Negative
If D < 0, the quadratic equation has two different complex roots. This is because, when D < 0, the roots are given by x = \(\dfrac{b \pm \sqrt{\text { Negative number }}}{2 a}\) and the square root of a negative number leads to an imaginary number always. For example \(\sqrt{4}\) = 2i. So when the discriminant of a quadratic equation is less than 0, it has two roots which are distinct and complex numbers (nonreal).
If Discriminant is Equal to Zero
If D = 0, the quadratic equation has two equal real roots. In other words, when D = 0, the quadratic equation has only one real root. This is because, when D = 0, the roots are given by x = \(\dfrac{b \pm \sqrt{\text { 0 }}}{2 a}\) and the square root of a 0 is 0. Then the equation turns into x = b/2a which is only one number. So when the discriminant of a quadratic equation is zero, it has only one real root.
A root is nothing but the xcoordinate of the xintercept of the quadratic function. The graph of a quadratic function in each of these 3 cases can be as follows.
Important Notes on Discriminant:

The discriminant of a quadratic equation ax^{2 }+ bx + c = 0 is Δ OR D = b^{2 }− 4ac.
 A quadratic equation with discriminant D has:
(i) two unequal real roots when D > 0
(ii) only one real root when D = 0
(iii) no real roots or two complex roots when D < 0
Related Topics:
Discriminant Examples

Example 1: Find the discriminant of the following equation: √3x^{2 }+ 10x − 8√3 = 0.
Solution:
The given quadratic equation is √3x^{2 }+ 10x − 8√3 = 0. Comparing this with ax^{2} + bx + c = 0, we get a = √3, b = 10, and c = 8√3.
The quadratic discriminant formula is:
D = b^{2}  4ac
= (10)^{2}  4(√3)(8√3)
= 100 + 96
= 196Answer: The discriminant = 196.

Example 2: Determine whether each of the following quadratic equation has two real roots, one real root, or no real roots. (a) 3x^{2 }− 5x − 7 = 0 (b) 2x^{2 }+ 3x + 3=0.
Solution:
(a) By comparing the given equation with ax^{2} + bx + c = 0, we get a = 3, b = 5, and c = 7. Its discriminant is,
D = b^{2}  4ac
= (5)^{2}  4(3)(7)
= 25 + 84
= 109So we have got a positive discriminant and hence the given equation has two real roots.
(b) By comparing the given equation with ax^{2} + bx + c = 0, we get a = 2, b = 3, and c = 3. Its discriminant is,
D = b^{2}  4ac
= (3)^{2}  4(2)(3)
= 9  24
= 15So we have got a negative discriminant and hence the given equation has no real roots.
Answer: (a) Two real roots (b) No real roots.

Example 3: What is the discriminant of quadratic equation 9z^{2 }− 6b^{2}z − (a^{4 }− b^{4}) = 0.
Solution:
By comparing the given equation with ax^{2} + bx + c = 0, we get a = 9, b = 6b^{2}, and c =  (a^{4 }− b^{4}). Its discriminant is,
D = b^{2}  4ac
= (6b^{2})^{2}  4 (9) [(a^{4 }− b^{4})]
= 36b^{4} + 36a^{4}  36b^{4}
= 36a^{4}Answer: The discriminant of the given quadratic equation is 36a^{4}.
FAQs on Discriminant
What is Discriminant Meaning?
The discriminant in math is defined for polynomials and it is a function of coefficients of polynomials. It tells the nature of roots or in other words, it discriminates the roots. For example, the discriminant of a quadratic equation is used to find:
 How many roots it has?
 Whether the roots are real or nonreal?
What is Discriminant Formula?
There re different discriminant formulas for different polynomials:

The discriminant of a quadratic equation ax^{2 }+ bx + c = 0 is Δ OR D = b^{2 }− 4ac.

The discriminant of a cubic equation ax^{3 }+ bx^{2 }+ cx + d = 0 is Δ or D = b^{2}c^{2 }− 4ac^{3 }− 4b^{3}d − 27a^{2}d^{2 }+ 18abcd.
How to Calculate the Discriminant of a Quadratic Equation?
To calculate the discriminant of a quadratic equation:
 Identify a, b, and c by comparing the given equation with ax^{2 }+ bx + c = 0.
 Substitute the values in the discriminant formula D = b^{2 }− 4ac.
What if Discriminant = 0?
If the discriminant of a quadratic equation ax^{2 }+ bx + c = 0 is 0 (i.e., if b^{2}  4ac = 0), then the quadratic formula becomes x = b/2a and hence the quadratic equation has only one real root.
What Does Positive Discriminant Tell Us?
If the discriminant of a quadratic equation ax^{2 }+ bx + c = 0 is positive (i.e., if b^{2}  4ac > 0), then the quadratic formula becomes x = (b ± √(positive number) ) / 2a and hence the quadratic equation has only two real and distinct roots.
What Does Negative Discriminant Tell Us?
If the discriminant of a quadratic equation ax^{2 }+ bx + c = 0 is negative (i.e., if b^{2}  4ac < 0), then the quadratic formula becomes x = (b ± √(negative number) ) / 2a and hence the quadratic equation has only two complex and distinct roots.
What is the Formula for Discriminant of Cubic Equation?
A cubic equation is of the form ax^{3 }+ bx^{2 }+ cx + d = 0 and its discriminant is in terms of its coefficients which is given by the formula D = b^{2}c^{2 }− 4ac^{3 }− 4b^{3}d − 27a^{2}d^{2 }+ 18abcd.
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