 Hi so again an ugly looking equation in front of us how to solve this big task isn't it but obviously we have some tools in our disposal one is or the most important one is we know how to solve quadratic equation but does this equation look like if I can use all of them it is it does how because you see if you notice there is a pattern there is something hidden so again mathematics all about observation and my observation says there is something which is you know some some trend is being observed now if I do something about it then if I use this particular observation then maybe I can solve this equation so let us try to first reduce it to some simpler looking form so can I say let's say y is equal to 3x square minus 4x if I if I do this it will become y plus the root y minus 6 is equal to 18 now it is much simpler than this one but still there is a third there's a radical which is creating a trouble how can I remove radical so radical the best way to kill radical is to square it but let's say if in this form if I square both sides what will happen I will this will be a plus b form and then again there will be a term if you square this up and open let's say I'm doing this you square this up you will again end up having a term which will contain this particular radical why because it will be simply y square plus root of y minus 6 whole square plus 2 times y times root y minus x isn't it this is a plus b plus a square plus b square plus 2 a b form but again this is not getting eliminated here this root will go this route will go but here again it will come back so hence I don't want this so can I do some trick the trick is very simple and that is if you let this radical stay alone on one side so hence let me just take away all this yep so hence there is no square anything anymore now what do I am doing this can I write this equation as let us keep this under root y minus x isolated and then what will happen it will become 18 minus 5 now if you square it will help yes why because now there is no more 2 a b term on the left hand side so let's square squaring both sides squaring both squaring both sides what will you get guys you will get under root y minus 6 squared whole square is nothing but 18 minus y whole squared isn't it now this will give you some peace yes so here you'll get 18 square minus 2 times 18 times y plus y square isn't it which is nothing but if you write this is and you know keep everything on same side so hence you'll get 18 square is 324 and then it is how much 2 minus 36 y and then I'm bringing this y on the right hand side you'll get minus 5 1 more minus 5 and then minus 6 goes on to this side becomes plus 6 right and then there is y square anyways and now flip everything or take everything on the left hand side you'll get y square now this plus this is minus 37 y and this is 324 plus 6 is 330 isn't it so I can write this as plus plus or minus so plus right plus 330 is equal to 0 clear so now it is reduced to finding the solution to this equation so can I split 330 if you see 330 is equal to 3 times 10 times 11 isn't it 30 times 11 correct and I have to reach 37 somehow so I have to break it into so it's also it's nothing but 3 into 3 not 3 into 2 rather 3 into 2 into 5 into 11 so do I see something okay so 37 how do I reach 37 22 and 15 22 and 15 will definitely help so hence what I'm saying is 22 is where is 22 this is 22 and 3 times 5 is 15 so hence it is 22 plus 15 is 37 and 22 into 15 is 330 so that will work so hence I'll write y square minus let's say 22 y minus 15 y plus 330 equals 0 hence y common y minus 22 minus 15 common y minus 22 is 0 so hence it is y minus 22 times y minus 15 equals 0 so y is equal to 22 and y is equal to 15 right this is what we'll get but we never wanted y we wanted x so what is y by the way so y was nothing but 3x square minus 4x if you see here 3x square minus 4x so hence again we'll get two equations 3x square minus 4x is equal to 22 and 3x square minus 4x is equal to 15 let's solve it so hence it is 3x square minus 4x minus 22 equals 0 and 3x square minus 4x minus 15 equals 0 again we'll go by the spreading the middle term so 3 times 22 is 66 which is also 11 times 6 but 11 times 6 will not give you 4 so 66 let's let me do the rough calculation here so I'm what I'm doing is you'd have guess by now 3 times 22 is 66 now 66 can be written as 2 into 3 is 6 into 11 right now 11 times 6 is 66 and what else 22 times 3 so looks a little ugly here but let us keep trying so maybe this or rather instead of breaking your head into middle terms and everything what you can do is you can straight away use the quadratic formula so here what will you get you will get x is equal to what will be x equal to minus b that is 4 plus minus under root b square b square is 16 minus 4 square is 16 minus 4 ac so minus 4 into 3 into minus 22 divided by twice of a which is 6 so if you solve it you'll get 4 plus minus under root 16 plus 12 times 22 so it is 16 plus 4 is 12 12 times 22 is 24 264 right divided by 6 right which is 4 plus minus under root 280 upon 6 okay now which is nothing but you can take it as 4 plus minus 280 can be written as 4 times 70 yep 4 times 70 and there is no possibility any further reduction so this is 6 so hence it will be 4 plus minus 2 root 70 so you can take this 4 out of the radical sign it will become 6 4 plus minus 2 root 70 upon 6 which is equal to 2 plus minus root 70 upon 6 sorry upon 3 yes so this is one set of a solution so this is one set of solution 2 plus minus root 70 upon 3 other set can be from here so how to find out 3 x square minus 4 x minus 15 so 3 15 3 times minus 15 is 45 which is also 9 times 5 and hence it will work it will work here so hence you'll see 3 x square can be written as minus 9 x plus 5 x minus 15 is equal to 0 so you take 3 x common you'll get x minus 3 you take 5 common you'll get x minus 3 equals 0 so hence it is x minus 3 times 3 x plus 5 equals 0 so hence you'll get x is either 3 or x is minus 5 upon 3 okay minus 5 upon 3 so these are the four solution to this equation guys x equals to 3 x equals to minus 5 upon 3 and these two right now this is what so what is the learning guys learning is first do not panic and then get exposed to as many different types of equations as possible and you all always learn that if this if this equation has to be solved it has to be solved with known methods and what is the only known methods so far so we know only solving quadratic equations and linear equations so hence all the equations must be thought towards or let's say it must be driven towards a form where you can apply your knowledge of quadratic equation or if you know how to solve these equations with some other method you can adopt that but at this level we have the knowledge of quadratic equation we'll try to resolve our given equation in form of quadratic equation