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Derive Quadratic Formula by Completing the Square

Introduction to Completing the Square:

In this article let us learn what is a quadratic equation and how to find the quadratic formula by completing the square.

An equation which is of the form ax2+bx+c=0 where a is not equal to zero is called as quadratic equation. For some values of x, the quadratic equation value becomes 0. The value at which the quadratic equation value becomes zero, is called as roots of the equation. For a quadratic equation there exists two roots. They can be real or imaginary. There are many methods to find the roots of the equation. One of the easiest method to obtain the roots of the equation is by quadratic formula. Let us derive the quadratic formula to find the roots of the quadratic equation by using method of completing squares.

Related Calculators
Completing Square Formula
 

Algorithm to Derive Quadratic Formula:

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Step 1: Consider the quadratic equation ax2+bx+c=0 where a is not equal to zero

Step 2: See whether a = 1. Suppose if a is not equal to one, then divide the equation throughout by the coefficient of x2

So ,ax2+bx+c=0 becomes x2+`(b)/(a)` x +`(c)/(a)` = 0

Step 3: The last term c/a is the constant term. Shift the constant term to the right hand side to get

                         x2 +`(b)/(a)` x = -`(c)/(a)`

Step 4: Find the half of co efficient of x, and square them.

              Half of coefficient of x is `(b)/(2a)` and square of it is `((b)/(2a))^2`

Step 5: Add the above term `((b)/(2a))^2` on both sides.

              x2 + 2`(b)/(2a)` x +`((b)/(2a))^2` = `((b)/(2a))^2` - `(c)/(a)`

Step 6 : Comparing the left hand side with a2 +2ab + b2 = (a+b)2, we could write teh above equation as

                (x +  `(b)/(2a)` )2 = `(b^2-4ac)/(4a^2)`

Step 7: Let us take positive square roots on both sides,

                 x + `(b)/(2a)` = `+-` `sqrt((b^2-4ac)/(4a^2))`

Step 8: As we need x, shift the constant term b/2a to the right hand side.

           So, x = -`(b)/(2a)` `+-` `sqrt((b^2-4ac)/(4a^2))` ----------------------- Quadratic formula

Deriving the Roots Using Quadratic Formula by Completing the Square

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Q;1 Solve the quadratic equation 9x2 -15x +6 = 0 by the method of completing the square.

9x2 -15x +6 = 0

Divide the equation throughout by the coefficient of xwhich is 9,

x2 - `(15)/(9)` + `(6)/(9)` = 0

x2 - `(5)/(3)` x  + `(2)/(3)` = 0

x2 - `(5)/(3)` x = - `(2)/(3)`

x2 - 2(`(5)/(6)` )x +`((5)/(6))^2` = `((5)/(6))^2` - `(2)/(3)`

 (x - `(5)/(6)` )2 = `(25 - 24)/(36)`

             = `(1)/(36)`

 x - `(5)/(6)` = `+-` `(1)/(6)`

 x = `(5)/(6)` `+-` `(1)/(6)`

x = 1 or x = 4/6 = 2/3


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