 Here we are taking up another question and in this question it is given that if the roots of equation x square plus 2cx plus ab equals to 0 are real and unequal prove that the given equation the other equation has no real roots. So that means certain conditions are to be met in the first given information basis that you have to prove the second part of the question correct. So the moment you see these words real and unequal what should come in your mind it should it is that whenever there are real and unequal roots to a given quadratic equation. So quadratic equation being x square 2cx plus ab equals 0 correct and we know that if there is a given quadratic equation ax square plus bx plus c equals 0 then d must be greater than 0 for real and unequal roots is it not. So this is what we have learned. So and what is d guys d is nothing but b square minus 4ac. Now this b, a and c all of them are in context of this equation so a is nothing but the coefficient of x square b is coefficient of x and c is the constant term so using the same principle here. Now don't get confused with whatever I have mentioned here as a b and c and whatever is given in this question the meaning of a b and c here is simply coefficient of x square coefficient of x and constant term respectively so we will deal with only those terms here as well. So if I write the discriminant of this given equation what will it be it is nothing but b square which is 4c square if you see b square is 4c square y let me you know write one more step so 2c square is b square here 2c square is coefficient of x here correct so 2c whole square minus 4 times coefficient of x square that is 1 multiplied by constant term that is a b and this has been given as this must be in fact this is the d and for this case since it is given that the roots are real and unequal so this must be greater than 0 correct. So from this equation in equation basically we get 4c square minus 4ab is greater than 0 now if there is an inequality if you divide the inequality by constant term the inequality a constant positive term the inequality doesn't change so what I am saying is I am writing here so dividing dividing dividing inequality by positive quantity doesn't impact the inequality what does it mean if you divide by a negative quantity then inequality will just change its character so hence here if you divide it by let's say minus 1 if you divide this entire inequality by minus 1 then this greater than equal to sign will become less than equal to but if you are dividing by positive quantity then the inequality doesn't change so hence I can cancel 4 out and I will get c square minus ab is greater than 0 so this particular information I have now c square minus ab is greater than 0 okay now let us take let us see what is the discriminant value of the second question so let us say this was d1 discriminant of the first and let us find out d2 discriminant of the second equation again if you see here what will it be minus 2a plus b whole squared see isn't it this is my b in the second second equation next minus 4 times a which is 1 again and c which is a square plus b square plus 2c square so what is this quantity what is this quantity let us simplify further so it will be 4 again minus 2 square is 4 and a plus b whole square and this will be minus 4 and a square plus b square plus 2c square now let us take 4 common then what will happen you will get a square plus b square plus twice ab isn't it because a plus b whole square is a square plus b square plus twice ab and this is minus a square minus b square and minus 2c square so I took 4 common so hence what is left behind a square b square 2ab and since here it will be minus 1 so because of that it is minus a square minus b square minus 2c square let's simplify further so this a square this a square will go this b square b square will go so what is left is 2 sorry 2ab minus 2c square and then you can see I can take one more 2 common so 4 into 2 and this becomes ab minus c square so it is 8 times ab minus c square now guys let us say this is 1 from 1 you can say from 1 we know that c square minus ab is greater than 0 okay so if you take these two terms on the right hand side what will you get 0 is greater than minus c square plus ab you will change the sign when you take it to the next other side or you can say ab minus c square is less than 0 ab minus c square is less than 0 and that's exactly what it is required right so hence ab minus c square is less than 0 then 8 times ab minus c square is also less than 0 and if you see guys these were all all these terms were d2 here equal to d2 isn't it so hence if you check this 8ab minus c square is 8ab minus c square which is nothing but d2 so d2 is less than 0 proved hence hence roots of our roots of the given quadratic equation roots of the equation are non real okay that's what it was asked to prove and hence we proved and what is the basis of the proof we could find out the discriminant value of the given equation in question and we could prove using the given information from the first equation that the discriminant of the second equation is less than 0 hence the result that's how you have to work it out so in any mathematics question what is given information try to extract some information from that given information and then always keep in mind what is that you know final achieve or let's say to be achieved goal and you try to bridge those two pieces of information and you get the result