 hi friends so welcome again to this problem solving session and continuing with our topic on nature of roots of a quadratic equation we have taken one problem over here and the problem says find the value of k k for which the equation has real and equal roots guys real and equal roots the moment we see this first thing we should come in our mind is the value of d so what do we know about nature of roots for you know the condition for nature of roots to be real and equal we know that discriminant must be 0 and discriminant is nothing but b square minus 4 a c where b is the coefficient of x is the coefficient of x square and c is constant term right so b is coefficient of x and a is coefficient of x squared and c is constant constant term so if we know that so for roots of a quadratic equation quadratic equation for roots of a quadratic equation to be equal to be equal we have we must have we must have to be equal and obviously real real and equal so I forgot I forgot this real and equal you must have d must be equal to 0 right that is b square minus 4 a c must be equal to 0 now let us deploy the values over here so if you notice carefully b is 3k here c is 4 and clearly a is 9 so the task is very simple now deploy the values so hence 3k whole square is b square minus 4 times a 9 times c that is 4 is equal to 0 so if I simplify this what will I get I'll get 9k square clearly and this is nothing but now minus 9 times 16 is 0 is it it so this implies if you see if you see 9 can be cancelled out isn't it 9 is common from both so hence it can be cancelled or you can say dividing the entire equation by 9 you'll get k square minus 16 is 0 or k square is equal to 16 or k is equal to plus minus root 16 right which is nothing but plus minus 4 so k could be either 4 or k is equal to minus 4 in both these cases there will be two cases that is k equals to 4 and k equals to minus 4 in both the cases roots will be roots of which equation the given equation the begin given equation was 9 x square plus 3k x plus 4 equals to 0 are real and equal okay this is what is the solution to this question let us take another question so if you see we have to find the value of k for which the given equation has real and equal roots so by now you already all of you know that you know for real and equal roots what is the criteria criteria is for let me write this for real and equal roots for real and equal roots of a quadratic equation we must have d which is nothing but b square minus 4 a c must be equal to 0 so this is the criteria for the real roots you know it okay so what are a b and c in this case if you clearly see a is one right one time x square it is right what is b guys b is minus 2 1 plus 3k so anything which is associated with the variable x is b and now c is 7 times 3 plus 2k isn't it which is nothing but 21 plus 14k right 7 21 plus 14k now let's you know complete the solution so let's find b these are much b square so b is minus 2 1 plus 3k and this is whole square and minus 4 times a is 1 and c c is 21 plus 14k isn't it right and this must be 0 so hence it is 4 times 1 plus 3k whole square minus 4 times 21 plus 14k must be equal to 0 so clearly this 4 is common to both so hence 4 can be eliminated right you can divide the entire equation by 4 and you will get you know the 4 does this particular 4 gets eliminated now you open the brackets you'll see this is something but 1 plus 2 times 1 times 3k plus 3k whole square isn't it a plus we whole square is a square plus 2 a b plus b square and this is minus 21 and this is minus 14k is equal to 0 right so minus 1 times 21 minus 21 and minus 1 times 14k is minus 14k here right so let's simplify further it is 1 plus 6k and this is 9k square so 3k whole square is 9k square many people make mistakes while doing these these simple you know mathematical manipulation so please be very very cautious and careful so minus 21 minus 14k many people I have seen they will do 3k square is 3k play 3k whole square they will write 3k square and here is where you get it wrong so be careful this is equal to 0 now what let's you know arrange them in an order so 9k square comes first 6k minus 14k is minus 8k okay and 1 minus 21 is minus 20 this is equal to 0 correct so what is it we get another quadratic equation in k so let us solve it by splitting the middle term so clearly it is 9 times minus 20 ac is 9 times minus 20 which is minus 180 guys so what is to be done so clearly 18 times 10 so minus 80 is also minus 18 times 10 so and why am I doing this because minus 18 plus 10 will be minus 8 that's what is the middle term splitting process right so hence I can now write 9k square minus 8k can be written as minus 18k plus 10k and why did I do this because 18 times 10 18 times 10 gives me 180 which was also equal to 9 times 20 so hence is it will help me in factorization this is the method for factorization so what can I do for the first two terms separate 9k take it common then what is left okay and what is left here to then what is a process just simply write k minus 2 and then complete it by writing 10 here because 10k will give you 10k here and 10 times minus 2 will give you minus 20 here so this becomes easier for me so hence this implies k minus 2 is a common factor and it is 9k plus 10 this is equal to 0 so our job is done right so what is what will this be solution will be either either k minus 2 is 0 or 9k plus 10 plus 10 is equal to 0 is it it so what will be the values like so the values will be so what will be the values like so the values will be k is equal to 2 or k is equal to minus 10 upon 9 okay so hence there are two values for k k equals to 2 or k equals k equals to 2 or k equals minus 10 upon 9 if this is so then the given quadratic equation will have real and equal roots right so this quadratic equation will have real and equal roots okay this is how you have to solve it so what did we do in this question we simply applied the condition this kermanant must be equal to 0 put the values of a b and c in a b and c there will be unknown k and with this condition d is equal to 0 we figured out what is the value of k and it came out to be k equals to 2 and k equals to minus 10 upon 9 right this is how you have to solve these kind of questions