Lesson THE INDIAN METHOD OF SOLVING A QUADRATIC EQUATION

The Indian Method of solving a quadratic equation is as follows:

step 1:
Move the constant term to the right side of the equation
step 2:
Multiply each term in the equation by four times the coefficient of the x^2 term
step 3:
Square the coefficient of the original x term and add it to both sides of the equation
step 4:
Take the square root of both sides
step 5:
Set the left side of the equation to the positive square root of the number on the right side and solve for x
step 6:
Set the left side of the equation to the negative square root of the number on the right side of the equation and solve for x

the method appears to be an offshoot of the completing the squares method for solving a quadratic equation.

it has an advantage over that method because it doesn't require you to make the coefficient of the x^2 term equal to 1.

i'll solve a problem for you using the method so you can see how it works.

PROBLEM

Find the roots of 6x^2 - x - 12

first thing you need to do is set the equation equal to 0.
this transforms the equation into the standard form of a quadratic equation.

your equation becomes:

6x^2 - x - 12 = 0

now we follow the steps of the indian method as shown below:

step 1:
Move the constant term to the right side of the equation

equation becomes:

6x^2 - x = 12


step 2:
Multiply each term in the equation by four times the coefficient of the original x^2 term

the coefficient of the original x^2 term is equal to 6.
multiply it by 4 to get 24.
multiply both sides of your equation from step 1 by 24 to get:

6x^2 - x = 12 becomes:

144x^2 - 24x = 288

step 3:
Square the coefficient of the original x term and add it to both sides of the equation

the coefficient of the original x term is equal to -1.
square it to get 1
add it to both sides of the equation from step 2 to get:

144x^2 - 24x = 288 becomes:

144x^1 - 24x + 1 = 289

step 4:
Take the square root of both sides

to find the square root factor, do the following.

take the square root of the coefficient of the x^2 term of the equation in step 3 to get:

square root of 144 = 12.

that will be the coefficient of the x term in your factor.

take the square root of the constant term of the equation in step 3 to get:

square root of 1 = 1.

that will be the constant term of your factor.

whether you will plus or minus the constant term to the x term in your factor is dependent on the sign of the middle term of the equation in step 3.

in this case, the middle term is negative, so your factor will be:

(12x - 1)^2 = 289

if you square your factor, you will get (12x - 1) * (12x - 1) = 144x^2 - 12x - 12x + 1 which is equal to 144x^2 - 24x + 1 which is the expression on the left side of the equation in step 3 so that's an indication that your factor is good.

you now have (12x - 1)^2 = 289

take the square root of both sides of this equation to get:

12x - 1 = +/- 17
add 1 to both sides of this equation to get:

12x = (+/- 17) + 1

step 5:
Set the left side of the equation to the positive square root of the number on the right side and solve for x

you get:
12x = 17 + 1 becomes:

12x = 18

divide both sides of this equation by 12 to get:
x = 18/12 which reduces to:

x = 3/2

step 6:
Set the left side of the equation to the negative square root of the number on the right side of the equation and solve for x

you get:
12x = -17 + 1 becomes:

12x = -16

divide both sides of this equation by 12 to get:
x = -16/12 which reduces to:

x = -4/3

your solutions to this quadratic equation are:

x = 3/2
x = -4/3

i know that these solutions are good because i created the original quadratic equation using the factors of:

(2x - 3) * (3x + 4) = 0

multiply these factors out and you get:
6x^2 + 8x - 9x - 12 = 0 which becomes:

6x^2 - x - 12 = 0

when you set (2x - 3) equal to 0, you get:
2x - 3 = 0
add 3 to both sides of the equation to get:
2x = 3
divide both sides of the equation to get:

x = 3/2

when you set (3x + 4) equal to 0, you get:
3x + 4 = 0
subtract 4 from both sides of the equation to get:
3x = -4
divide both sides of the equation by 3 to get:

x = -4/3

you can also confirm that these value are good by solving the original equation using those values of x.

when x = 3/2, the original equation of 6x^2 - x - 12 = 0 becomes 6(3/2)^2 - (3/2) - 12 = 0 which becomes 6(9/4) - (3/2) - 12 = 0 which becomes (54/4) - (3/2) - 12 = 0 which becomes (27/2) - 3/2) - 12 = 0 which becomes (24/2) - 12 = 0 which becomes 12 - 12 = 0 which becomes 0 = 0.

this confirms the value of x = 3/2 is good.

when x = (-4/3), the original equation of 6x^2 - x - 12 = 0 becomes 6(-4/3)^2 - (-4/3) - 12 = 0 which becomes 6(16/9) - (-4/3) - 12 = 0 which becomes 2(16/3) - (-4/3) - 12 = 0 which becomes (32/3) + (4/3) - 12 = 0 which becomes (36/3) - 12 = 0 which becomes 12 - 12 = 0 which becomes 0 = 0.

this confirms the value of x = (-4/3) is good.

you could also solve have solved this quadratic equation by using the quadratic formula or by using the completing the squares method or by using the factoring method.

all methods arrive at the same answer.

the method used in this lesson is called the Indian method.

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