 Hi friends, so we have seen how to solve equations which are not really in the form of quadratic equation by using the methods and you know rules of solving quadratic equations. So here again if you see we have a deadly looking equation and it is you know in the first glimpse it is looking as if it is appearing as if we are not going to solve it but we can use some trick and some pattern recognition to arrive at a form which can be easily solved using the rules of solving a quadratic equation. Now there are few things which are to be taken care of so if you see some some observation so mathematics again as I have been saying it's all about observation so there is seven x here can you see this seven x. Now if you remember any polynomial quadratic polynomial breaks down in the form of let's say if if the coefficient of x square is one then it will be something of this sort x minus alpha times x minus beta isn't it where alpha beta alpha plus beta will be the sum alpha plus beta will be nothing but some let's say the coefficient of x so here seven is alpha plus beta so if you see seven can be broken down into two plus five or three plus four right so with this hint this hint I would be proceeding further so hence I will be clubbing these two terms because three plus four ends up being seven and two and five also is seven so let's try and see what happens so if I write this as x plus two times x plus five times x plus three times x plus four and in the right hand side it is 24 x square plus seven x plus seven right now if you see on the left hand side the if you know you just multiply these two so what will happen it will be x square plus two x plus five x plus 10 first and then in second two will be x square plus three x plus four x plus 12 right and this is equal to 24 times x square plus seven x plus seven now this is a catch if you see if you have noticed by now this is x square plus seven x plus 10 right so I am getting x square plus seven x here similarly in the second term also you're getting x square plus seven x plus 12 here also you're getting x square plus seven x and anyways you had x square plus seven x here as well now friends it is much more easier to solve such kind of problems why because I see a trend I can utilize this trend so what is this what can I do I can say let y be equal to x square plus seven x right and then reduce this into this form y plus 10 into y plus 12 in is equal to 24 y plus seven now this is within our reach we will be able to solve such equations why because if you now expand you will get what will you get you will get an equation in y and that is a quadratic equation so hence expand it you will get y square plus 10 y plus 12 y plus 120 is equal to 24 y plus 24 7 to 168 right yes verified so hence what now you know simplify simply so y square and then 10 y plus 12 y is 22 y minus 24 y is minus 2 y and this will be reduced to minus 48 is equal to zero so can you see we started from we are an ugly looking equation very very you know frightening equation but now it appears to be very simplistic very beautiful equation which is inviting to solve it okay so let's solve it so hence what is it I can again use the spreading the middle term so 48 is 24 times 2 so I can or 24 times 2 will not work it's 8 times 6 rather so y square minus let's say 8 y plus 6 y minus 48 equals 0 so this implies y is common so y minus 8 and here it is 6 common so y minus 8 is equal to 0 that means y minus 8 and y plus 6 is 0 that means what y is either it or y is equal to minus 6 correct so here we got y but we this question didn't demand y it demanded x so what was y let's see y was this x square plus 7 x so that means we got two quadratic equations okay so too much work x square plus so this becomes just a mundane mechanical work to solve this quadratic equation the only catch was the real mathematics was involved here where you had to think how to convert this equation into something which is beyond or let's say within our reach so first equation is this and the second equation is clearly x square plus 7 x equals minus 6 let's solve both of them it should be a cake work now 7 x minus 8 equals to 0 and this is x square x square plus 7 x plus 6 equals to 0 is it so hence it is x square now you can write this as plus 8x minus x minus 8 equals to 0 or for this you can write x square plus x plus 6x plus 6 equals 0 spreading the middle term always becomes helpful so x times x plus 8 minus x sorry minus 1 times x plus 8 equals 0 and hence this is nothing but x times x plus 1 and 6 times sorry so I should not be writing like that it should be plus 6 times x plus 1 equals 0 that means this is nothing but x plus 8 times x minus 1 equals 0 and this becomes x plus 1 times x plus 6 equals 0 so mechanical isn't it now once you have cracked the real problem other things just is a formality so hence from here it x is either minus 8 or x is equal to 1 these are two solutions from here or from here you'll get x equals to minus 1 or x equals to minus 6 so these are the four solutions minus 8 1 minus 1 6 you can check by deploying it back into the quadratic oh sorry whatever initial equation was given and you will see all of them satisfy the given equation that's how we convert ugly looking terrible looking equations into something which is very familiar so be you know because after quadratic or let's say there is an equation which is of degree 3 or more you rest assured that you have to convert it into some beautiful looking equation to solve it further right and there must be something something hidden in the problem itself which will help you to reduce or convert this convert the equation from the existing form to a quadratic or even linear form okay so keep that in your mind