Discriminant Formula: Definition, Formula, Problems

Discriminant Formula: In mathematics, the discriminant is a function of the polynomial coefficients. A polynomial’s discriminant is a number that depends on the coefficients and affects certain properties of the roots in mathematics. It’s usually described as a polynomial function of the original polynomial’s coefficients. In polynomial factoring, number theory, and algebraic geometry, the discriminant is commonly employed. The symbol \(D\) or \(∆\) is frequently used to represent it.

Discriminant Formula: What is a Quadratic Equation?

An algebraic statement of the second degree in \(x\) is called a quadratic equation. A quadratic equation has the standard form \(a{x^2} + bx + c = 0,\) where a and b are coefficients, \(x\) is the variable, and \(c\) is the constant term. The coefficient of \({x^2}\) is a non-zero term \((a≠0),\) is the first criterion for determining whether an equation is quadratic.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is \(a{x^2} + bx + c = 0.\)
Where \(a→\) coefficient of \({x^2},b \to \) coefficient of \(x, c→\) constant.

Quadratic Equation Formula

The quadratic formula is the most efficient way to determine the roots of a quadratic equation. Some quadratic equations are difficult to factorise. In these cases, we can apply this quadratic formula to find the roots. The sum of the roots and the product of the roots of the quadratic equation can also be found using the roots of the quadratic equation. 

The quadratic formula is given by

\(x = \frac{{ – b \pm \sqrt {{b^2} – 4ac} }}{{2a}}\)

The above formula for getting the roots of a quadratic equation is known as the quadratic formula.

Roots of the Quadratic Equations

The two values of \(x\) obtained by solving the quadratic equation are the roots of a quadratic equation. The symbols alpha \(\left( \alpha \right)\) and beta \(\left( \beta \right)\) are used to represent the roots of a quadratic equation. The zeros of the equation are also known as the roots of the quadratic equation. 

Nature of the Roots of the Quadratic Equations

The nature of the roots of a quadratic equation can be determined without actually finding the problem’s roots \((α, β)\). The discriminant value, which is part of the formula for solving the quadratic equation, help us understand the nature of roots. The discriminant of a quadratic equation is denoted as \(D\) and is equal to \({b^2} – 4ac.\) The discriminant value determines the nature of the quadratic equation’s roots.

What is Discriminant?

A quadratic equation’s number of solutions is determined using the discriminant formula. The discriminant is the name given to the expression in the quadratic formula that comes under the square root (radical) sign.

When solving quadratic equations, the term \({b^2} – 4ac\) is used.

It can “discriminate” between the following categories of responses:

1. We get two real solutions when it’s positive.
2. When it is zero, there is only one real solution (both answers are the same)
3. When it is negative, there are two complex solutions.

Formula of Discriminant

A polynomial’s discriminant is a function of its coefficients and is denoted by a capital \(“D”\) or the delta symbol \(∆.\) It demonstrates the nature of the roots of any quadratic equation with rational coefficients \(a, b,\) and \(c.\) A quadratic equation can simply indicate the real roots or the number of \(x-\)intercepts. This formula is used to determine if the quadratic equation’s roots are real or imaginary.

We know \({b^2} – 4ac\) determines whether the quadratic equation \(a{x^2} + bx + c = 0\) has real roots or not, \({b^2} – 4ac\) is known as the discriminant of these quadratic equations.

Therefore, a quadratic equation \(a{x^2} + bx + c = 0\) has

1. Two distinct real roots, if \({b^2} – 4ac > 0\)
2. Two equal real roots, if \({b^2} – 4ac = 0\)
3. No real roots, if \({b^2} – 4ac < 0\)

Example of Discriminant

Q.1. Find the discriminant of the equation \(2{x^2} – 6x + 3 = 0,\) and hence find the nature of its roots.
Ans: From the given equation, we obtain \(a = 2,b = – 6,c = 3\)
Discriminant \({b^2} – 4ac = {( – 6)^2} – 4 \times 2 \times 3\)
\( = 36 – 24\)
\( = 12 > 0\)
Therefore, the given equation has two distinct real roots.

Symbol of Discriminant

The symbol used to represent discriminant is either \(D\) or \({\rm{\Delta (delta)}}{\rm{.}}\)

\(D\) or \(\Delta = {b^2} – 4ac\)

Graphical Representation of Discriminant Formula

1. If \({b^2} – 4ac > 0\) or \(D>0\) means the given equation has \(2\) real roots. The \(x-\)axis of the graph is crossed at two points.
2. If \({b^2} – 4ac = 0\) or \(D=0\) means the given equation has \(1\) real root. The \(x-\)axis of the graph is crossed at one point.
3. If \({b^2} – 4ac < 0\) or \(D<0\) means the given equation has \(0\) real roots. The \(x-\)axis of the graph is not crossed.

Learn the Concepts of Quadratic Equations

Uses of Discriminant

1. The discriminant is the part of the quadratic formula underneath the square root symbol \({b^2} – 4ac.\) The discriminant tells us whether there are two solutions, one solution, or no solutions.
2. The discriminant of a quadratic equation is significant because it indicates the number and the nature of solutions. This knowledge is very important since it acts as a double check when applying any of the four ways to solve quadratic equations (factoring, completing the square, using square roots, and using the quadratic formula).

Solved Examples – Discriminant Formula

Q.1. Find the discriminant of the quadratic equation \(3{x^2} – 2x + \frac{1}{3} = 0\) and hence find the nature of its roots.
Ans: From the given \(a = 3,b = – 2,c = \frac{1}{3}\)
Discriminant \({b^2} – 4ac = \left( { – {2^2}} \right) – 4 \times 3 \times \frac{1}{3}\)
\(=4-4=0\)
So, for the given quadratic equation, two equal real roots exist. Or, we say only one real root exists.

Q.2. Find the discriminant of quadratic equation \({x^2} + 7x – 60 = 0,\) and hence find the nature of its roots.
Ans: From the given equation, we get \(a=1, b=7, c=-60\)
Discriminant \({b^2} – 4ac = \left( {{7^2}} \right) – 4 \times 1 \times ( – 60)\)
\(=49+240\)
\(=289>0\)
So, for the given quadratic equation, two real roots exist.

Q.3. Find the discriminant of quadratic equation \(2{x^2} – 4x + 3 = 0,\) and hence find the nature of its roots.
Ans: From the given \(a=2, b=-4, c=3\)
Discriminant \({b^2} – 4ac = \left( { – {4^2}} \right) – 4 \times 2 \times (3)\)
\(=16-24\)
\(=-8<0\)
Therefore, the given quadratic equation has no real roots.

Q.4. Find the discriminant of quadratic equation \(2{x^2} – 3x + 5 = 0,\) and hence find the nature of its roots.
Ans: From the given \(a=2, b=-3, c=5\)
Discriminant \({b^2} – 4ac = \left( { – {3^2}} \right) – 4 \times 2 \times (5)\)
\(=9-40\)
\(=-31<0\)
So, for the given equation, no real roots exist.

Q.5. Find the discriminant of quadratic equation \(3{x^2} – 4\sqrt 3 x + 4 = 0,\) and hence find the nature of its roots.
Ans: From the given \(a = 3,b = – 4\sqrt 3 ,c = 4\)
Discriminant \({b^2} – 4ac = {( – 4\sqrt 3 )^2} – 4 \times 3 \times 4\)
\(=16×3-48\)
\(=48-48\)
\(=0\)
Therefore, the given equation has two equal real roots.

Q.6. Find the discriminant of equation \(6{x^2} + 10x – 1 = 0,\) and hence find the nature of its roots.
Ans: From the given equation, we obtain \(a=6, b=10, c=-1\)
Discriminant \({b^2} – 4ac = \left( {{{10}^2}} \right) – 4 \times 6 \times ( – 1)\)
\(=100+24\)
\(=124>0\)
So, the given quadratic equation has two real roots.

Q.7. Find the discriminant of the equation \({x^2} – 2x + 3 = 0,\) and hence find the nature of its roots.
Ans: From the given \(a=1, b=-2, c=3\)
Discriminant \({b^2} – 4ac = \left( { – {2^2}} \right) – 4 \times 1 \times 3\)
\(=4-12\)
\(=-8<0\)
So, for the given quadratic equation, no real roots exist.

Summary

The discriminant is the part of a quadratic formula underneath the square root symbol \({b^2} – 4ac\) The discriminant tells us whether there are two solutions, one solution, or no solutions. This article includes the definition of quadratic equation, standard form of a quadratic equation, quadratic equation formula, roots of the equation, nature of the roots, discriminant definition, formula of discriminant, symbol of discriminant, and its uses.

This article helps in better understanding the topic discriminant formula. This article’s outcome helps us learn the discriminant formula and to solve various problems based on it.

Frequently Asked Questions (FAQs) – Discriminant Formula

The most commonly raised queries on Discriminant Formula are answered here:

We hope this detailed article on the discriminant formula helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!