 Hello friends, welcome again to problem solving session on algebraic identity. So if you see the previous sessions we are dealing with algebraic, algebraic identities and we have discussed time and again that is a very important and very vital area in mathematics. Now the given question is prove that 2a square plus 2v square plus 2 square minus twice a b twice b c times twice c a will be equal to this will be equal to this. So if you see both sides we have square terms and clearly if there are square terms we have learned few identities which involve squaring and that's a plus b whole square and then we have a minus b whole square and then we have a square minus b square these are the expansions we already know. So if you see there is a twice a b term here there is a twice b c term and twice c a so this is inviting why because twice a b and that to minus that mean if I have an a square and b square I can complete the square. So with that indication I can see there are 2 a squares b b require only one to complete the square isn't it. So let me take one square from here so I can write a square and then one b square I can take from here so I can write b square and then club this with this term minus 2 a b. So if you see I have got one full square now and what about others so then I have minus 2vc here so if I see do I have an extra b square yes we have so I will write b square here now if you see this b square plus b square b square will give you this 2b square so that's done so now I take one c square here out of these two c squares and write it here then finally the third term will be minus 2ca so I take one c square and one a square here right. So now if you see in the in the given expression this a square plus this a square becomes 2a square that's fine this c square and this c square becomes another 2c square that's also there and this b square and this b square we have already taken 2b square and the rest three terms as it is but the beautiful part was this now is going to give give us something and what is that this is nothing but if you look closely a minus b whole square right similarly this b square plus c square minus 2bc term is nothing but b minus c whole square and this term is nothing but c minus a whole square which is equal to if you see RHS isn't it guys so very simple if you know algebraic identities you can prove these kind of algebraic expressions okay in this question we have to find out the coefficient of x square term in the expansion of this given algebraic expression okay now what is a coefficient coefficient you as you know is the constant term attached to that particular term so for example in this certainly there will be terms containing x to the power 4x squared and some constant terms isn't it these will be the terms in this and x cube as well so these will be you know and x as well so these are the terms which will be there in this expansion right and we have to find out the coefficient of x square here okay how to do it so obviously expanding where you use this identity a plus b plus c whole square as a square plus b square plus c square plus twice a b plus twice bc plus twice c a but then it looks like it is too much of a work so can we just try and even that and use some other method yes in the realm of algebraic identities as well there are lots of solutions now if you see what we can do is we can do some clubbing here so let us say this particular thing here can be written as x square plus 1 plus x whole square is it and the other term is plus x square plus 1 minus x whole square so we have just rearranged these two terms so what is the advantage you will see this looks like same the first two terms are same in both the square terms here isn't it so if I club them together and then what do I see so if you see if I treat this as a and this as b and this is a and this is b what do I get I get and if I open the square using the identity which identity a plus b whole square is equal to a square plus 2 a b plus b square what do I get I get x square plus 1 whole squared then 2 times x squared plus 1 times x then plus x squared then the second term will be nothing but x square plus 1 whole squared minus 2 x square plus 1 times x plus x squared isn't it so if you see this term and this term goes cancels out right so what do I get I get 2 times x square plus 1 whole square plus 2 x square now expanding this is much easier so if you see this is nothing but 2 times x to the power 4 plus 2 x squared plus 1 plus 2 x squared so expansion of this is this one hence the final answer 2 x to the power 4 plus 4 x square plus 2 and then 2 x square so if we club all of them like terms together you will get 2 x to the power 4 plus 6 x square plus 2 now what is the coefficient of x square in this guys clearly 6 is the answer so answer equals 6 okay so instead of using a trinomial expansion we used only binomial that is only 2 terms square expansion and we could save some time