 Hello and welcome to the session. In this session, first of all let us discuss special rules of quadratic equations. Now we know that the standard form of the quadratic equation is AX squared plus BX plus C is equal to 0 where A is not equal to 0 and ABC are the constants. Now let P and Q be the roots of this equation and also we know that the sum of the roots that is P plus Q is equal to minus the coefficient of X over the coefficient of X squared in the given equation. So this is equal to minus Q over L and the product of the roots that is P2 is equal to the absolute term over the coefficient of X squared in the given equation which is C over A. Now first of all let us discuss reciprocal roots. Now if one of the roots that is P is equal to 1 by Q that means one root is the reciprocal of the other root. So this implies PQ is equal to 1. Now P2 is the product of the roots which is equal to 0 over A so this implies C over A is equal to 1 therefore we have C is equal to A. That means the coefficient of X squared is equal to the constant term in the given equation the roots are reciprocal roots. Now let us discuss 0 roots. Now let us discuss the case 1. Now if one root of the given equation is equal to 0 now let P be equal to 0 then the product PQ will be also equal to 0. Product PQ is equal to C over A is equal to 0 therefore C is equal to 0 that is the constant term is equal to 0. Now in the case 2 if both the roots are equal to 0 let us let P is equal to Q is equal to 0 then in that case the product of the roots that is PQ is equal to 0 which implies now the sum of the roots and the product of the roots is C over A is equal to 0 therefore B is equal to C is equal to C short the given equation I equal to 0 then in that case and the absolute term R equal to 0 are the roots the quadratic equation AX squared plus BX plus C is equal to 0 let this be equation number 1 also and be found by these two roots and the equation will be that is x whole into x plus product of the roots is equal to 0. The equation of the root is equal to 1 then B is equal to minus of P plus Q the whole which is further equal to minus B which is equal to P plus Q that is 1 by P and 1 by Q. Now for these two roots we can again power equation and that will be equal to x square minus 1 by Q the whole into A into 1 by Q that is equal to 1 by minus equal to C equal to minus equal to 1 so putting all these values here plus BX plus A is equal to the case 1 of infinite roots that is 1 is infinity. Now we are let infinity then it is 1 by P will be equal to 0. Now from the case 1 of 0 roots if 1 of 0 roots of the given equation is equal to 0 then of equation number if 1 by P is equal to 0 that is 1 of the root of this equation is equal to 0 then the constant term that is A will be equal to 0 of this equation. Now P is the root of equation number 1 therefore the corresponding root of equation number 1 is infinity so we have A is equal to 0 in my equation now let us discuss the case 2 when both of the roots of the given equation are infinity now we are let P and Q then in this case the reciprocal of the roots that is 1 by P will be equal to 0 and 1 by Q will be equal to 0. Now from the case 2 of 0 roots if both of these of the given equation are 0 then in that case the constant term constant term which is A will be equal to 0 and B will be equal to 0 equal to 0 and B is equal to 0 in equation number 1. Thus the conditions are if 1 root of the given equation is infinite 2 to 0 that is coefficient of x square is equal to 0 if both of the roots of the given equation are coefficient of x now both roots will be positive then the sum of the root the product of the root which implies minus B over A positive A and B is negative and the product is positive thus implies minus B over A is negative the root which is C over A is positive A will be negative even the product will be positive if A, B and C are all of same signs that is root should be negative when A, B and C are all of same signs. Now the roots will be of opposite signs negative that is 0 that is but minus B over A is equal to 0 when B is equal to 0. Let us find the condition when the 2 product equation we have plus B over x plus C over A is equal to 0 and A 2 x square plus B 2 x plus C 2 is equal to 0 where A 1 is not equal to 0 and A 2 is not equal to 0 and A 1 B 1 C 1 and A 2 B 2 and C 2 are the constants. Now let alpha be the common root of the given product equations it means it will satisfy the given quadratic equations. So we have A 1 alpha square plus B 1 alpha plus C 1 is equal to 0 and A 2 alpha square plus B 2 alpha plus C 2 is equal to 0. Now solving by the method of cross multiplication B 1 C 2 minus B 2 C 1 is equal to alpha over is equal to 2 minus A 2 B 1. Now equating the first 2 B 1 is equal to B 1 C 2 minus B 2 C 1 is equal to A 1 C 2 is equal to A 1 B 1. Now let this B equation A and this B equation B. Now equating A and B minus B 2 C 1 is equal to C 1 A 2 minus A 1 C 2 A 2 B 1. That is both of these are the values of alpha. So we have equating this and find the root minus B 2 C 1 the whole and B 2 minus A 2 B 1 the whole is equal to which is the required condition for the two quadratic equations to have one common root. Now let us find the condition, check equations. We have both roots common. Now again these two quadratic equations add this B equation X and this B equation Y. Now let alpha and beta are the common root equations X and Y. Beta is equal to minus B 1 over A 1 and the product of the roots that is alpha beta is equal to C 1 over A 1. And from the equation X that is alpha plus beta would be equal to minus B 2 over A 2 and the product of the roots that is alpha beta is equal to C 2 over A 2. Now both of these that is the sum of alpha and beta. Therefore we have minus B 1 over A 1 is equal to minus B 2 over A 2. These are the values of C 1 over A 1 is equal to C 2 over A 2. Now from the first one B 2 is equal to A 1 over 2 B 1 over B 2 is equal to C 1 over C 2 the required condition to take equations to have both common root. And now let us discuss what is an identity. Now an identity a mathematical relationship an example identical to X square plus 2 A X plus A square. Now here is used to distinguish it from the sign of an equation. Now let us discuss an important property of identities and that is in an identity the coefficients of the life paths of the variables in world are equal and both the sides. That is into X raised to power n plus A n minus 1 into X raised to power n minus 1 plus so on into X raised to power 2 plus A 1 X plus A naught is identical to B n into X raised to power n plus B n minus 1 into X raised to power n minus 1 plus so on plus B 2 into X raised to power 2 plus B 1 X plus B naught between X is equal to 0 in this identity we get X is equal to B naught. Now canceling A naught B naught from both the sides and dividing throughout by X we get n into X raised to power n minus 1 plus A n minus 1 into X raised to power n minus 2 plus so on plus A 2 X plus A 1 is identical to B n into X raised to power n minus 1 plus B n minus 1 into X raised to power n minus 2 plus so on up to B 2 X plus B 1. Now again in this identity putting X is equal to 0 B naught A 1 is equal to B 1 and then repeating A 2 is equal to B 2 A 3 is equal to B 3 etcetera to B n A n minus 1 is equal to B n minus 1 up to A 2 is equal to B 2 A 1 is equal to B 1 and A naught is equal to B naught. Hence we can say in an identity of coefficients of light powers of the values involved are equal at both the sides. So in this session you have learnt about special roots common root and identity. So this completes our session hope you all have enjoyed the session.