✅ Discriminant Formula ⭐️⭐️⭐️⭐️⭐

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 ax² + bx + c = 0 is in terms of its coefficients a, b, and c. i.e.,

• Δ OR D = b² − 4ac

Do you recall using b² − 4ac earlier? Yes, it is a part of the quadratic

Comparing the equation with ax³ + bx² + cx + d = 0, we have a = 1, b = 0, c = -3, and d = 2. So its discriminant is,

Δ or D = b²c² − 4ac³ − 4b³d − 27a²d² + 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 ax² + 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 x-coordinate of the x-intercept 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 ax² + bx + c = 0 is Δ OR D = b² − 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

What is Discriminant Formula?

How to Calculate the Discriminant of a Quadratic Equation?

What if Discriminant = 0?

What Does Positive Discriminant Tell Us?

What Does Negative Discriminant Tell Us?

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 – b²-4ac. 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 = ax² + bx + c, and the discriminant of a quadratic equation, in this case, is b² − 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 ax² + 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 x² – 2x + 3 = 0? Also, determine the number of solutions this equation has.

Solution:

Given, x² – 2x + 3 = 0

In the equation,

a = 1 ; b = -2 ; c = 3

The formula for discriminant is,

Δ = b² – 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: b² − 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: b² − 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: b² − 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: b² − 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 (b² − 4ac) is placed at the denominator 
• Keep an eye out for negative b². 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:

✅ Math Formulas ⭐️⭐️⭐️⭐️⭐