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Roots of a Quadratic Equation Methods, Formulas, Applications and Calculator

 

A quadratic equation is an equation of degree 2 in the form ax²+bx+c = 0, where a is not equal to 0. The value of x in this equation is called the roots of the quadratic equation. There are only two roots in a quadratic equation. The nature of these roots can be real and imaginary.

Nature of roots of a quadratic equation

Value of discriminant Nature of roots
D > 0 Real, distinct
  D is a perfect square Rational roots
  D is not a perfect square Irrational roots
D = 0 Real, equal
D < 0 Complex, distinct (a pair of complex conjugates)

Where D is called the discriminant of the root and is equal to b² – 4ac.

The universal equation to find roots of a quadratic equation ax² + bx + c = 0 is given by:

image

Methods to find the roots of a quadratic equation

1. Factorization method: - It is the simplest method to find the roots of a quadratic equation. It is not applicable to all forms of the quadratic equation but is suitable for simple forms.

Example

Find the roots of the quadratic equation x² – 6x + 5 = 0 by factorization method.

Solution

By trial and error, we need to find the factors that make 6x and can be taken common from the terms x² and 5.

The above equation can be written as x² – 5x – x + 5 = 0

This implies, x (x-5) – 1 (x-5) = 0

This gives, (x-5) (x-1) = 0

We get the roots as x = 1 or x = 5.

2. Quadratic formula: - We can use the universal quadratic formula to find the roots of every quadratic equation.

image2

Example

Find the roots of the equation x² – 6x + 5 = 0 from the quadratic formula.

Solution

Compare the given equation with the equation ax² + bx + c = 0.

We get, a = 1, b = -6, c = 5

By putting these values in the universal equation, we get

image2

We get, x = 5 or x = 1, which are the roots of the quadratic equation x² – 6x + 5.

Fun facts

  • The formula to find the two roots of a quadratic equation was found by a Sanskrit philosopher, Sudhara.
  • Completing the square to resolve common issues with areas was introduced by the Babylonians in 400 BC.
  • Pythagoras was the mathematician to find the first reliable mathematical quadratic formula in 500 BC.

Applications

  1. A quadratic equation is used to find trajectory motions and their paths in physics.
  2. They are used to calculate areas under a curve and the speed of an object if it traces a circular path.
  3. It is used in sports and athletics to maximize the results and bring out profitable outcomes.
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