 Hi friends so again you know we are here back with another question and in this question it is said that if PQR are real and P is not equal to Q then show that the roots of the equation P minus Q x square plus P times P plus Q minus 2 times P minus Q equals 0 are real and unequal okay so the keywords are real real and unequal isn't it these are the two keywords so what do I know what do we know about this so if the roots have to be real and unequal then what is the condition so we know for any quadratic equation if roots are for roots to be real and unequal okay condition is D is equal to B square minus 4 AC must be greater than 0 what are B what is B what is A what is C so we know that if I and B and B A and C are nothing but coefficients of the variables here so A x square plus B x plus C equals 0 so this was the quadratic equation if this quadratic equation has to have real and unequal roots then this condition must be true so let us find out in the given if so our given equation is this one so what are the values of A B and C so I will write it out like this a is here is P minus Q isn't it B is clearly five times oh sorry I have missed an x here yep correct so this is x so hence B will be equal to five times P plus Q and C is equal to minus 2 P minus Q correct so let us find out what is the B what is the value of D here in this case so D here is five square P plus Q whole squared minus 4 A is P minus Q and C is minus 2 P minus Q right so hence it is nothing but 25 P plus Q whole square I'm purposefully not opening this bracket I'll tell you why I did that and minus minus in this the second term is eight times P minus Q whole square right now if you see guys 25 is a positive quantity now any square is yeah greater than zero why P plus Q whole square is greater than zero why because squares are always always greater than I'll write greater than equal to because it can be zero also and always greater than equal to zero but it's given it was given if you see what was given P is not equal to Q okay so we'll be using this particular information here now in the second term you see P minus Q is greater than zero I'm not writing greater than equal to zero why because P is not equal to Q given so definitely P and P is not equal to Q then P minus Q is not equal to zero so definitely P minus Q whole square will be greater than zero because squares are never negative okay that means 25 times P plus Q whole square which is a positive point greater than zero this is clearly greater than equal to zero if P was equal to minus Q then it will become zero and eight times P minus Q whole square is certainly greater than zero now if you add both of them what will you get you will get 25 P plus Q whole square plus something which is greater than zero I'm adding something which is greater than zero so clearly it will be greater than zero it will not be equal to zero why because this quantity is always greater than zero so hence the total I'm adding something to it so as total will always be greater than zero right that means what D in this case is greater than zero and this sufficient condition for real and unequal roots guys okay so the given equation this given equation will have will always have real and unequal roots