# Discriminant

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Quadratic means a variable that is multiplied by itself. The operation essentially includes squaring. A general quadratic equation is –

ax2 + bx + c = 0

With the help of this formula, the roots of a quadratic equation can be found. This root pertains to the value represented by ‘x’.

### What is Discriminant Quadratic Function?

x = $\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$

Within this quadratic formula, the discriminant function is found under a quadratic formula. It is represented as b²-4ac and the discriminant can be zero or positive or negative. It indicates whether there will be no solution, one solution, or two solutions.

The formula of discriminant algebra exhibits the following characteristics -

• When discriminant is zero, it shows that there are repeated real number solution to the quadratic;

• For a negative discriminant, neither of the solutions amount to real numbers;

• For a positive discriminant, there are two distinct real number solutions to the quadratic equation.

The following quadratic equation would show a number of solutions, which would illustrate to the students to find the discriminant.

For example, the given quadratic equation is –

6x2 + 10x – 1 = 0

From the above equation, it can be seen that:

a = 6,

b = 10,

c = -1

Applying the numbers in discriminant –

b2 − 4ac

= 102 – 4 (6) (-1)

= 100 + 24

= 124

Given that, the discriminant amounts to be a positive number, there are two solutions to the quadratic equation.

### 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

For a detailed understanding of discriminant, you may avail of online classes or PDF materials of Vedantu.

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.