Discriminant of a Quadratic Equation Explained

In the case of quadratic equations, the discriminant is commonly employed to determine the nature of the roots. Though determining a discriminant for any polynomial is difficult, we may use formulas to get the discriminant of quadratic and cubic equations.

In arithmetic, a polynomial's discriminant is a function of the polynomial's coefficients. It's useful for figuring out what kind of solutions a polynomial equation has without having to locate them. The name "discriminant" comes from the fact that it distinguishes between the equation's solutions (as equal and unequal; real and nonreal).

It is commonly denoted by Δ or D. The discriminant's value can be any real number (i.e., either positive, negative, or 0).

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’. 

Formula and Relationship between Roots and Discriminant

Any polynomial's discriminant (Δ or D) is defined in terms of its coefficients. The discriminant formulas for a cubic equation and a quadratic equation are:

Discriminant formula of a quadratic equation:

ax2  bx + c = 0 is

Δ or D = b2 − 4ac

Discriminant formula of a cubic equation:

ax + bx³ + cx² + d = 0 is

Δ or D = b2c2 − 4ac3 − 4b3d −27a2d2 + 18abcd

Relationship between Roots and Discriminant

The values of x that satisfy the equation are known as the roots of the quadratic equation ax2 + bx + c = 0. 

To find them, use the quadratic formula:

X = \[\frac{-b\pm \sqrt{D}}{2a}\]

Although we cannot discover the roots using the discriminant alone, we can determine the nature of the roots in the following way.

If discriminant is positive:

There are two real roots to the quadratic equation if

D > 0. 

This is because the roots of D > 0 are provided by x =

\[\frac{-b\pm \sqrt{\textrm{Positive number}}}{2a}\]

And a real number is always the square root of a positive number.

When the discriminant of a quadratic equation exceeds 0, it has two separate and real-number roots.

If discriminant is negative:

The quadratic equation has two different complex roots if

D < 0. 

This is because the roots of D < 0 are provided by x =

\[\frac{-b\pm \sqrt{\textrm{Negative number}}}{2a}\]

and so when you take the square root of a negative number, you always get an imaginary number.

If discriminant is equal to zero:

The quadratic equation has two equal real roots if D = 0. 

This is because the roots of D = 0 are provided by x = \[\frac{-b\pm \sqrt{0}}{2a}\]

and 0 would be the square root. The equation thus becomes x = −b/2a, which is a single number. When a quadratic equation's discriminant is zero, it has only one real root.

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. 

Things to Remember While Using Quadratic Formula 

• It is absolutely necessary that the arrangement of the equation is made in a correct manner, else the solution cannot be obtained. 

• 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.