Quick review of the quadratic formula
The quadratic formula says that
What is the discriminant?
The discriminant can be positive, zero, or negative, and this determines how many solutions there are to the given quadratic equation.
 A positive discriminant indicates that the quadratic has two distinct real number solutions.
 A discriminant of zero indicates that the quadratic has a repeated real number solution.
 A negative discriminant indicates that neither of the solutions are real numbers.
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.
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
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 =
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 =
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
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
= 196
Answer: 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
= 109
So 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
= 15
So 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.
Why is Discriminant Formula Important?
Using the discriminant, the number of roots of a quadratic equation can be determined. A discriminant can be either positive, negative or zero. By knowing the value of a determinant, the nature of roots can be determined as follows:
 If the discriminant value is positive, the quadratic equation has two real and distinct solutions.
 If the discriminant value is zero, the quadratic equation has only one solution or two real and equal solutions.
 If the discriminant value is negative, the quadratic equation has no real solutions.
1. When the Discriminant is Zero, How Many Numbers of Solution Would a Quadratic Equation Have?
Ans. At the outset, the discriminant or determinant of a quadratic equation is a component under the square root of the quadratic formula – b^{2}4ac. If the discriminant is equal to zero, then there can be only one unique solution. If discriminant amounts to less than zero, no solution will arise. However, if it is more than zero, there can be two real solutions to an equation.
2. What are the Various Forms of a Quadratic Equation?
Ans. The three main forms of a quadratic equation are – (1) Standard form, (2) Factored form, and (3) Vertex form.
The standard form of a quadratic equation is represented as y = ax^{2} + bx + c, and the discriminant of a quadratic equation, in this case, is b^{2} − 4ac. The Factored form of quadratic equation is represented as y = (ax + c) (bx + d). The Vertex form of quadratic equation is represented as y = a (x + b)^{2} + c.
It must be noted that a, b, and c numbers.
3. What is the Significance of Quadratic Equations?
Ans. The application of quadratic equation is found in our everyday lives ranging from calculation of areas, determination of profit of a product, or formulation of an object’s speed.
There will be at least one squared variable within the quadratic equation and represented in the form of ax^{2} + bx + c = 0, where x is the variable, ‘a’ ‘b’ ‘c’ are constants and ‘a’ does not amount to zero. Using the discriminant quadratic function, the solution to the equation can be found out.
Example Question Using Discriminant Formula
Question 1: What is the discriminant of the equation x^{2} – 2x + 3 = 0? Also, determine the number of solutions this equation has.
Solution:
Given, x^{2} – 2x + 3 = 0
In the equation,
a = 1 ; b = 2 ; c = 3
The formula for discriminant is,
Δ = b^{2} – 4ac
=> Δ = (2)^{2 }– 4(1)(3)
=>Δ = 4 – 12
Δ = 8 < 0Since the value of the determinant is negative, the equation will have no real solutions.
Relationship Between Roots and Discriminant
The relationship between discriminant and roots can be understood from the following cases –
 Case 1: b^{2 }− 4ac is greater than 0
Here,
a, b, c = real numbers
a ≠ 0
discriminant = positive
Then, the roots of the quadratic equation are real and unequal.
 Case 2: b^{2 }− 4ac is equal to 0
Here,
a, b, c = real numbers
a ≠ 0
discriminant = zero
Then, the roots of the quadratic equation are real and equal.
 Case 3: b^{2 }− 4ac is less than 0
Here,
a, b, c = real numbers
a ≠ 0
discriminant = negative
Then, the roots of the quadratic equation are not real and unequal. In this instance, the roots amount to be imaginary
 Case 4: b^{2 }− 4ac is greater than 0 as a perfect square as well
Here,
a, b, c = real numbers
a ≠ 0
discriminant = positive and perfect square
Then, the roots of the quadratic equation are unequal, real, and rational.
Things to Remember While Using Quadratic Formula
 It is absolutely necessary that the arrangement of the equation is made in a correct manner, else solution cannot be arrived at
 Ensure that 2a and the square root of the entire (b^{2} − 4ac) is placed at the denominator
 Keep an eye out for negative b^{2}. Since it cannot be negative, be sure to change it to positive. The square of either positive or negative will always be positive
 Retain the +/. Watch out for two solutions
 While using a calculator, the number will have to be rounded on a specific number of decimal places
Quadratics with Irrational Solutions
Solution(s) to quadratic equations won’t always be “nice” round numbers.
For example, we’ll be seeing below, the quadratic equation:
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