Roots of Quadratic Equation by Quadratic Formula

A quadratic equation is a two-degree quadratic equation and it has two a maximum of two roots. They can either be real, or imaginary. 

The roots of the quadratic equation is the value that satisfies the quadratic equation thus, we can say that if the given quadratic equation is, ax² + bx + c = 0, and if α is a root of the quadratic equation, then α satisfies the quadratic equation, i.e., aα² + bα + c = 0. We also define roots of the quadratic equations ax² + bx + c =  as the zeros of the polynomial ax² + bx + c.

We can easily find the roots of the quadratic equation by using the following quadratic formula,

[Tex]\bold{x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}} [/Tex]

This gives the two roots of the quadratic equation, 

Taking +ve Sign

x = [-b + √(b² – 4ac)] / 2a

Taking -ve Sign

x = [-b – √(b² – 4ac)] / 2a

Thus using the quadratic formula we easily get the roots of the quadratic equation.

Discriminant of a Quadratic Equation

The discriminant of a quadratic equation ax² + bx + c = 0 is denoted by D and given by,

D = b² – 4ac

Discriminant of a Quadratic Equation is very helpful in determining the nature of the root of quadratic equations.

Nature of Root of Quadratic Equation

To find the nature of the roots of the quadratic equation we find the discriminant of the given quadratic equation. The term is called discriminant because it determines the nature of the roots of the quadratic equation based on its sign.

There are 3 types in the nature of roots, 

• Real and distinct roots: For real and distinct roots, the discriminant should be positive i.e. b² – 4ac > 0. The curve of the equation intersects the x-axis at two different points.
• Real and equal roots: For real and equal roots, the discriminant is zero i.e. b² – 4ac = 0. The curve of the equation intersects the x-axis at only one point.
• Complex roots: For complex roots, the discriminant is negative i.e. b² – 4ac = 0. The curve of the equation does not intersect the x-axis.

Maximum and Minimum Value of Quadratic Expression

The maximum and minimum values for the quadratic equation of the form ax² + bx + c = 0 can be observed with the help of graphs.

• If the value of a is positive i.e. (a > 0), the quadratic equation has a minimum value at x = -b/2a i.e., -D/4a.
• If the value of a is negative i.e. (a < 0), the quadratic equation has a maximum value at x = -b/2a i.e., -D/4a.

Where D is the discriminant of the Quadratic Expression.

Quadratic Formula is used to find the roots (solutions) of any quadratic equation. Using the Quadratic formula real and imaginary all the types of roots of the quadratic equations are found.

The quadratic formula was formulated by a famous Indian mathematician Shreedhara Acharya, hence it is also called Shreedhara Acharya’s Formula. It is used to find the solution of the quadratic equation of the form ax² + bx + c = 0. So, let’s start learning about the concept of Quadratic Formula.