 Hello friends welcome again to this problem solving session. I hope you are keeping a good pace of problem solving because to ace mathematics you need to solve as many problems as possible. So continuing with our practice of solving problems this is another question I have come up with. So this question says if a plus b plus c is equal to 6 the value of a plus b plus c is given and bc plus ca plus ab value is also given that is 11 you have to find the value of aq plus bq plus cq minus 3abc. The moment you see this you are you know the first thing which should come to your mind is the identity special identity which we learned and that is aq plus bq plus cq minus 3abc is equal to what first a plus b plus c this is the first factor and multiplied with the second factor which is nothing but a square plus b square plus c square minus minus ab minus bc minus ca is it not. So hence very clearly we have been given ab plus bc plus ca the value has already been given. So now we have to just figure out what will be the value of aq plus bq plus cq minus 3abc right okay fair enough now in this question we are going to utilize one more identity now if you see closely in this particular problem we know the value of a plus b plus c isn't it that is 6 but we do not know the value of a square plus b square plus c square minus ab minus bc minus ca is also known why because if bc or I'm writing it like ab plus bc plus ca is equal to 11 then minus ab minus bc so minus ab and minus bc and minus ca will be equal to negative 11 simply so this is known to us but one thing which we don't know is what is the value of this particular term a square plus b square plus c square if I somehow find out a square plus b square plus c square then my job is done I will simply deploy all the values here and we'll get this value isn't it so but how to find out a square plus b square plus c square now again if a plus b plus c is known do I not know you know there's an identity which is related to this which is nothing but a plus b plus c whole squared is what you know already it is nothing but a square plus b square plus c square plus 2 ab plus 2 bc plus 2 ca isn't it we know this already now we are going to use this identity to find the value of a square plus b square plus c square which is required in this step okay so tell me what will it be so I can write a plus b plus c whole squared minus 2 ab minus 2 bc minus 2 ca will be equal to a square plus b square plus c square I simply did what I simply took all this to this left hand side okay and rearranging you can you can now write what can you say about a square plus b square plus c square it is nothing but so hence I write a square plus b square plus c square is equal to a plus b plus c whole squared and if you see I can take minus 2 common out of these three isn't it this three I can take minus 2 common I can write ab plus bc plus ca now thank god we have been given these values isn't it now a plus b plus c was given to be equal to 6 c here and bc plus ca plus ab or ab plus bc plus ca whichever way you want to say it's 11 so it is nothing but 6 squared minus 2 times 11 so which is nothing but 36 minus 22 which is equal to 14 isn't it so hence we got a square plus b square plus c square is equal to 14 now what we have to just deploy all the values here so hence now we know that a q plus b q plus c q minus 3 abc is equal to a plus b plus c which value is anyways known which is 6 and then a square plus b square plus c square value we just found out which is 14 minus ab plus bc plus a right I have taken minus sign common minus sign common so hence this value is nothing but 11 so just deploy it is nothing but 6 into 14 minus 11 which is equal to 6 into 3 which is equal to 18 right so see using the knowledge of algebraic identity is we could solve this problem and let's revisit once again so what was told what was given a plus b plus c was given that is 6 bc plus ca plus ab was given that is 11 you have to find out a q plus b q plus c q minus 3 abc how did we do that we know that a q plus b q plus c q minus 3 abc equals abc a plus b plus c times a square plus b square plus c square minus ab minus bc minus ca this one we knew these values two of the three values were known a plus b plus c was known this was known but this was not known so hence we used another identity which one here it is and using this identity we could figure out what is the value of a square plus b square plus c square and then finally we deployed the value in the given identity here and we found the value to be equal to 18 so in the same problem keep in mind that you might need to use more than one type of identity