derive shridhar acharya formula by completing square method
Answers
Step-by-step explanation:
If we have to solve any quadratic equation, we would probably switch to factorization method. But in some cases, it is not convenient to solve quadratic equations by factorization method.
For example, consider the equation x2 + 4x + 2 = 0. In order to solve this equation by factorization method we will have to split the coefficient of the middle term 4 into two integers whose sum is 4 and product is 2. Clearly, this is not possible in integers. Therefore, we cannot solve this equation by using factorization method.
We can convert any quadratic equation to the form (x + a)2 – b2 = 0 and can easily find its roots. Let us consider the equation 3x2 – 5x + 2 = 0. Here coefficient of x2 is not a perfect square. So, we multiply the equation throughout by 3 to get 9x2 – 15x + 6 = 0
Now we have, 9x2 – 15x + 6
= (3x)2 – 2 x (3x) x 5/2 + 6
= (3x)2 – 2 x (3x) x 5/2 + (5/2)2 – (5/2)2 + 6
= (3x-5/2)2 – 25/4 + 6
= (3x-5/2)2 – ¼
(3x-5/2)2 = ¼
3x-5/2 = 1/2 or 3x-5/2 = -1/2
3x=1/2+5/2 or 3x=-1/2+5/2
3x=6/2 or 3x = -4/2
x=3/3 or x=-2/3
x=1 or x = -2/3
Here is a formula for finding the roots of any quadratic. It is proved by completing the square. In other words, the quadratic formula completes the square for us. This method is popularly known as Sridharacharya’s formula as it was first given by an ancient Indian mathematics Shridharacharya around 1025 A.D
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