 Hi friends, I hope you're doing good. So let's continue with our problem solving sessions now You would have seen this kind of problems Where in you have to simplify 2x plus 5 plus 3 and 2x plus 5 y plus 4 so there are two product terms So there are two factors basically and you have to find out the product now one way is you know that you can You can do this expansion what expansion you can start with 2x times 2x plus 5 y plus 4 plus 5 y times 2x plus 5 y plus 4 and then finally Plus 3 times 2x plus 5 y plus For this is a usual method and you know this is a usual algebraic expansion if you see but then it will take a lot of time Is it it is it so can we reduce reduce this you know or simplify this process hence You know, let us try to let us try to find another method by which our efforts are reduced So mathematics friends is always, you know about finding out trends and patterns if you see the pattern here, there is 2x plus 5 y and 2x plus 5 y here as well. So there are 2x plus 5 y in both the Terms, so let me just say let a be equal to 2x plus 5 y Okay, so what will happen if you now see these two factors these are a plus 3 and a plus 4 Isn't it now this looks familiar because we have learned an identity Wearing and let me just now throw some light on the identity which we have learned So x plus a times x plus b this is the algebraic identity We have learned and this is nothing, but if you if you know, this is nothing but x square plus a Plus b times x Plus a b. This is what we have learned now here clearly instead of x. We have a here Isn't it this is a this is a this is instead of x and instead of you know This a we have 3 and here b is 4 so hence using the same identity We can say this is nothing but a squared plus 3 plus 4 times x sorry not x here is a So hence I have to write a and plus 3 into 4 That means this is a square plus 7 a plus 12 a square plus 7 a plus 12 so it's you know while solving It's also a good practice to keep in mind that kind of expressions you are getting So in in later chapters, you will see that this particular expression will be given and you will be asked to factorize it See, so I'm not asking you to mug it up, but then keep these things in mind. So what expressions are you getting by? You know Let's say multiplying two factors so that when if you get a problem where you have to factorize the given expression Then it becomes easier for you right now. We know already what is a so let us substitute that to x plus 5 y This goes for square and then 7 times 2 x plus 5 y and Then plus 12 right now. This is much easier if you see this is again We can use an identity for the first term which one you can you have it up guessed it by now So a plus we all whole square is a square It's 2 a b plus B square so in here it will be nothing but 2 x whole squared plus 2 times 2 x times 5 y now these dots are not decimals. These are my formal duplication and then 5 y whole squared Is it it then what 7 times 2 x is 14 x plus 35 y and then plus 12 So hence this is nothing but 4 x squared 2 into 2 2 square is 4 and x square is x square and then here if you simplify the second term you'll get 4 into 5 20 x y and This is 25 Y square isn't it then what? 14 x plus 35 y Plus 12 so this is my final answer because there are no like terms. So I can't really simplify it Further so hence the answer is 4 x square plus 20 x y plus 25 y square plus 14 x plus 35 y plus 12 so this is how you have to use identity algebraic identities to simplify the Given expression has been given that x plus 1 by x is 6 So this is given and the basis this you have to find out x square plus 1 plus x square x square plus 1 by x square and x to the power 4 plus 1 by x to the power 4 So again, please keep this thing in mind that whenever you see such kind of a Expression so if you see you know some linear term is here and you are expected to find out the square value So one thing which should come to your head is Algebraic identity so if you see x plus 1 by x is 6 Is it it now? I know can I square both sides? So I'm writing squaring squaring both sites Squaring both sides only then I can expect an x square on the left hand side Isn't it so hence it will be x plus 1 by x whole square is equal to six square So now if you see this is again and this can be expanded using an identity which is a plus b Whole square is a square plus twice a b plus B square Let us use this. So if you see this will become x square plus two times x times one upon x and Plus one upon x whole Squared is equal to 36 Now here is the catch guys this two times x times 1 by x will get reduced to two only isn't it? So hence this particular problem becomes interesting. So if you see it is nothing but x squared plus This one upon x whole square can be written as 1 square by x square. So hence simply 1 by x square, isn't it? Plus 2 y because if you see this x and this x gets cancelled out if x is not equal to 0 So this x and this x gets cancelled out this one because x is not equal to 0 so hence So after cancelling out you can see this is x square plus 1 by x square plus 2 is equal to 36 This implies if I take this two on the other hand on the side, sorry You'll get 36 minus 2 which is 34 Right. So what do I get x square plus 1 x square is 34? Now the second part of the question says talks about x bar 4 So clearly if I I need power 4 here So how to get it again you square the term square the expression you'll get it isn't it? So let us say x square Plus 1 by x squared now on squaring this is equal to 34 Squared okay Now this is nothing but x again. You use the a plus b whole square. I need you will get x to the power square to 2 power 2 plus 2 times x square times 1 upon x square plus 1 upon x squared whole squared is equal to 34 square now 34 square guys is 1156 Okay, now if you simplify this further you'll get x to the power 4 plus 2 Plus 1 upon x to the power 4 and this is equal to 1 1 5 6 So that means x to the power 4 Plus 1 by x to the power 4 will be 1156 minus 2 is equal to 1 1 5 4 so this would be the answer