Paravartya Yojayet

 Paravartya Yojayet" is another Vedic Mathematics sutra that deals with solving quadratic equations. 

The phrase can be translated as "Transpose and adjust."

This sutra is used to solve quadratic equations of the form ax^2 + bx + c = 0, where 'a,' 'b,' and 'c' are constants, and 'x' is the variable you need to find.

The process involves four steps:

Transpose: Move the constant term (c) to the other side of the equation.

Adjust: Multiply 'a' and 'c,' and adjust the coefficient 'b' to make it twice the square root of the product of 'a' and 'c.'

Factorization: Factor the left-hand side of the equation into two binomial expressions.

Solve for 'x': Set each binomial expression equal to zero and solve for 'x.'

Let's go through an example to illustrate the process:

Example: Solve the quadratic equation 2x^2 + 5x + 3 = 0 using "Paravartya Yojayet."

Step 1: Transpose

Move the constant term (3) to the other side of the equation:

2x^2 + 5x = -3

Step 2: Adjust

Multiply 'a' and 'c': 2 * (-3) = -6

Now, adjust the coefficient 'b' to be twice the square root of the product of 'a' and 'c':

b = 2 * √(2 * -3) = 2 * √(-6)

Step 3: Factorization

Now, rewrite the left-hand side of the equation as two binomial expressions:

2x^2 + 2√(-6)x + 3x - 3 = 0

Factor the expressions:

2x(x + √(-6)) + 3(x + √(-6)) = 0

Step 4: Solve for 'x'

Set each binomial expression equal to zero and solve for 'x':

2x + √(-6) = 0

2x = -√(-6)

x = -√(-6) / 2

x + √(-6) = 0

x = -√(-6)

Remember that the square root of a negative number is represented as 'i' in complex numbers, where 'i' is the imaginary unit (i^2 = -1). So, the solutions to the quadratic equation are:

x = (-√6 / 2) - (√6)i and x = (-√6 / 2) + (√6)i.

Using the "Paravartya Yojayet" technique, you can efficiently solve quadratic equations and find both real and complex solutions.