 Hello friends, so we are going to take up yet another method of solving quadratic equations in this session. So now this rule is called Sridharacharya's rule, Sridharacharya's rule. Now who was Mr. Sridhara. So Sridharacharya as you know, as you can understand acharya means teacher. So Sridhara was, or Sridhara acharya let me, yeah, so Sridhara acharya was an Indian mathematician somewhere around 8th, 9th century AD. Okay, that time you know Indian mathematician as well as Arabic mathematicians were getting you know involved into studies of algebra. So in this regard, Sridhara acharya's work is pretty famous and today we call the same thing as quadratic formula as well, quadratic formula. So this formula is very, very useful to solve quadratic equations. So now let us understand how to solve a quadratic equation using Sridhara acharya's rule. It is actually if you see it's pretty close to the methods which we have already studied. So in the last session we studied completing the square method and more or less resembles the same. So let us start with the method. So let us say we have an equation ax square plus bx plus c equals 0. This is a quadratic equation and time and again we have learned a cannot be equal to 0 and a, b, c all belong to set of real numbers. Now first step is step number one, you multiply, multiply, multiply both sides by 4a that is 4 times the coefficient of x square. So 4 times coefficient of x square right in this case it is a so let us multiply it by 4a. We will get 4a x 4a square x square plus 4abx plus 4ac is equal to 0. Okay second step is subtract 4ac from both sides, both sides. So what will you get? You will get 4a square x square plus 4abx plus 4ac which was already there. Now I am subtracting 4ac from here and this is 0 minus 4ac. Okay then third step is add b square to both sides, both sides. What will it be? It will be simply now I am simplifying simultaneously so 4ax square plus 4abx and if you see these two terms will cancel each other out so it was 0 here but now I am adding b square to both sides so b square minus 4ac. Now all this method, all this step will help me completing the square. So hence complete the square on LHS. Okay so what is it? It will be 2ax whole square if you see closely plus 2 times 2ax times b plus b square is equal to b square minus 4ac. Correct? So the left hand side is nothing but 2ax plus b whole square is equal to b square minus 4ac. Okay and we can call it this is equal to 2ax plus b whole square is equal to d square where d is equal to... So basically yeah d or rather what we can say is let it be d. Yeah instead of d square let we call it as d. So what is d guys? d is equal to b square minus 4ac. You will later see this is called discriminant. Okay so this implies I can write this as 2ax plus b whole square minus root d whole square is equal to 0. So I took this capital D on the left hand side and d can always be written as root d whole square. I am writing this for a purpose because if you see so this resembles a square and this resembles b square. So hence I am seeing a difference of square terms here. So hence a square minus b square is how much a square minus b square is a minus b times a plus b isn't it? So we will use this thing here. So hence using that difference of square thing I can write. So I can write 2ax plus b minus root d times 2ax plus b plus b plus root d equals 0. So these are two separate areas yeah. Okay so hence again we have seen this how to solve such... So hence you can simplify also. So it is 2ax plus b minus root d and it is 2ax plus b plus root d. Okay this is equal to 0. So hence what will happen? Either 2ax plus b minus root d equals to 0 or 2ax plus b plus root d equals to 0. That means either 2ax is equal to minus b plus root d or 2ax is equal to minus b minus root d minus b minus root d or 2ax is equal to minus b minus root d. So hence hence from here you will get x is equal to minus b plus root d upon twice a or x is equal to minus b minus root d upon twice a. Right? So substituting the value of b you can write under plus b square minus 4ac upon twice a or x is equal to minus b minus root b square minus 4ac upon twice a. Isn't it? So this is how you... This is what we found out as the solution. So solution R 2 1 is x is equal to minus b plus under root b square minus 4ac by 2a and other one is this. The only difference between these two is between these sign here plus here one is plus another is minus. So there are two solutions of the given quadratic equation. That is what is called Sridhacharya's method. So once again what is the rule? Multiply both sides by 4a then subtract minus 4ac from both sides then add b square to both sides complete the square and factorize and solve.