 Hello and welcome to this session. In this session, we will discuss the banding of symmetric expressions of the roots. First of all, let us discuss what is a symmetric expression. Now an expression f of xy is said to be symmetric, f of xy is equal to f of yx. Now let us discuss an example. Now here, the expression f of xy is equal to x square plus 2x square y plus 2y square x plus y square and the expression f of yx is equal to y square plus 2y square x plus 2x square y plus x square. See that the expression f of xy, that is this expression and the expression f of yx, that is this expression, are equal to symmetric expression. These are the symmetric expressions of the roots. Now we know that the standard form of the quadratic equation is ax square plus bx plus c is equal to 0 where a is not equal to 0 and a, b, c are the constants, b are the roots. This is the equation and now we will find out the value of the functions and for this we will proceed with the following steps. Now the first step is the product of the function that is the given equation. Then in the second step, express the function in terms next step, substitute plus q and pq. From the given equation let it be equation number 1, the values of the functions of p and q, we do not find the values of p and q separately. So for finding out the values of the functions of p and q, we will only follow these steps. Now let us discuss the following relations, p plus q square, that is if we have to find out p square plus q square, then it can be written as pole square minus 2pq. Now the value of p plus q, that is the sum of the roots from the given equation and also pq, that is the product of the roots from the given equation and then substitute those values and this equation will get the value of p square plus q square. Now it can be written as pq. Now again we have expressed this relation in terms of p plus q and pq and substituting the values of p plus q and pq here, we will obtain the value of p minus q whole square. Here we will express the other relations also in terms of p plus q and pq, that is the next relation which can be written as p the whole, q the whole, into, now from the second relation it can be solved, which is equal to q the whole, p square minus p the whole into pq into p plus q in terms of p plus q. So by putting the values of the sum of the roots and product of the roots, value of pq plus qq. Now next we have to find pq minus qq. Now by the formula it can be written as p minus q the whole into p square plus pq plus q square the whole, which is further equal to e minus q the whole into p plus q whole square minus pq the whole, which is further equal to, this can be written as square root of e plus q whole square minus 4 pq into, now this relation is also expressed in terms of the sum and the product of the root. Now let us discuss q raise to power 4, which can be written as p square plus q square whole square minus 2 p square q square, which can be further written into pq p plus q and pq. Now the next relation is p raise to power 4 minus q raise to power 4, which can be written as q square the whole into p square to the formula for this it will be p plus q the whole into p minus q the whole into plus q square the whole equal to, now we have the value of p and from this we can find the value of p minus q. All these values here this will be equal to p plus q the whole into p minus q square root of the whole is also expressed in terms of p plus q and therefore whenever we have to find out the value of the roots in terms of p plus q that is the sum of the roots and pq that is the product of the roots and then by substituting the values of p plus q and pq we can find out the value of the given relation. Now let us discuss one example a quadratic equation x square minus 5x plus 4 is equal to 0 and if a and b are the roots of the given quadratic equation then we are doing symmetric expressions first we will see b plus, now for finding out the value of these symmetric expressions first of all we will find the sum and the product of the roots from the given equation. Now given the quadratic equation as x square minus 5x plus 4 is equal to 0 so here plus b as a and b are the roots of this equation is equal to minus x that is 1 so a plus b is equal to now the product of this is equal to a b is equal to 2 which is 4 oh which is 1 so this is equal to 4. Now in the first part we have to find a square plus b square now by using this first result square in terms of a plus b and a b so this is equal to a plus b whole square minus 2 a b. Now putting the values of a plus b and a b here this is equal to minus 2 into 4 which is equal to 25 minus 8 which is equal to 17. Start with the second part now again we have a b plus a b cube in terms of a plus b and a b. For this taking a common it will be a b into a square plus b square whole is equal to now a b is 4 into therefore this is how we can solve the values of this metric expressions of the roots. And now let us discuss formation of equivalent q are the root equal to q, b is equal to 0, b is equal to 0 this b the whole into x minus q the whole is equal to 0 or we can write it as q the whole into x not the roots therefore we have x is equal to p and x is equal to q are the roots of the equation. We have got x minus q is equal to 0 and x minus q is equal to 0 and for formula the equation we will multiply and form this we will get a quality equation in n that is x square minus now b plus q is the sum of the roots so it is the product of the root to find out an equation whose roots are given to us and product of the roots and then the required equation is roots the whole into x plus product of the roots is equal to 0. Now let us discuss it with the help of an example and this we have to find the equation whose roots are 2 and 3. Now find the solution the first step we will find the sum and the product of the roots so the sum of the roots and the product of the roots is the whole that is 5 into x that is plus 6 is equal to our equation whose roots are 2 and 3. Next let us learn how to find the condition when a relation between 2 roots. Now before starting this first of all let us review that for a quality equation a x square plus b x plus c is equal to 0 where a is not equal to 0 the sum of the roots is equal to minus b over a that is minus coefficient of x over coefficient of x square and product of the roots is equal to c over a that is 10 over coefficient of x. Now let us discuss the procedure to find the condition and the relation between 2 roots is given and in the second step write the other rule and condition that is the first rule which we have supposed in step 1 and the second rule which we will obtain by using the given condition. Now in the next step write the sum using the sum and the product of the roots from the given quality equation of an example this we have to find the condition that 1 root of a x square plus b x plus c is equal to 0 may be 3 times the other. Now let us start with the solution. Now in the first step let the quality equation be alpha. Now according to the given equation the condition is 1 to us that is 1 root 3 times the other root. So we have supposed 1 root as alpha root of the given quality equation will be equal to 3 alpha. Now for the given quality equation will be equal to minus b over a. Now we have supposed 1 root as alpha the quality equation is 3 alpha therefore sum of the roots that is alpha plus 3 alpha is equal to minus b over a is equal to minus b over a. So that is equal to minus b over 4 a. Now the product of the roots of the given quality equation is c over a roots alpha and 3 alpha of the given quality equation the product of the roots will be equal to alpha into 3 alpha which is equal to c over a which implies that is equal to c over a. Now put in the best implies 3 into minus b over 4 a whole square is equal to c over a which further implies that is equal to c over a we have eliminated alpha and that the required formula is equal to 16 a c which is the required relation right which is the required condition when 1 root of the given quality equation a square plus b x plus c is equal to 0 may be 3 times the other session we have learnt about the values of the symmetric expressions of the roots formation of equations as to find the condition when a relation between the two roots is given. For fashion hope you all have enjoyed the session.