 So, in this question, we have to find the values of k again, k again for which the equation x square plus 5kx plus 16 equals 0 has no real roots, no real roots. So, you know it by now for a quadratic equation, ax square plus bx plus c equals 0. If d is less than 0, then it is sufficient for no real roots, no real roots. And what is d guys? b is b square minus 4 ac, this is the condition. So, let us start with it. So, hence what is b? If you write, if you see here, a is equal to 1 clearly, if you match, this is a here and what is a here? 1, 5k happens to be the b, so 5k and c is clearly 16. Now what? Let us do this b square. So, b square is 5k square minus 4 times 1 times c, that is 16, this must be less than 0. 5k square is 25k square and this is 64 minus 64 is less than 0. So, how to find the values of k? So, 25k square must be less than 64 or 5k whole squared must be less than 8 squared. Now, how to solve such kind of inequalities? So, 5k is less than 8 square. So, two possibilities 5k, 5k can be negative or 5k can be positive. This 5k has, we do not have any restriction on k. So, k can be, let us say k is 2 cases, k is 1. Now, k can be less than 0, that is negative or k, second case could be k is greater than 0. Two possibilities are there. In that case, what will happen? 5k will be less than 8, in both the cases, 5k has to be less than, 5k square rather should be less than 8. So, how to find such, basically how to solve such inequalities? It is, at the end of the day, it is an inequality. So, how to find such inequality is this way. So, hence what you write is this, 5k whole square minus 8 square is less than 0. So, hence, 5k minus 8 and 5k plus 8 should be less than 0. It should be less than 0. That means there are two factors here and product of two factor is less than 0. That means, if 5k minus 8 is less than 0, then 5k plus 8 must be greater than 0. So, if you see, we have two inequalities here. That is, if I consider 5k minus 8 to be less than 0, then 5k plus 8 must be greater than 0. Then only the product of these two factors will be less than 0. Example, if I have, let's say, 6 and minus 8, if I multiply them, then only the product is less than 0. Otherwise, if both are negative and both are positive, then the product is always greater than 0. For example, minus 2 into minus 2 is 4, which is greater than 0. Similarly, 2 into 2 is 4, which is again greater than 0. So, hence, if something is, both the products, both the factors are positive or both the factors are negative, then the product is positive. Otherwise, if one is positive, one is negative, the product is negative. So, hence, by that logic, if 5k minus 8 is less than 0, then 5k plus 8 is greater than 0. Then you might have a question, then why am I considering, how do I know that this term or this factor is less than 0 and this is more than 0? I am saying, I don't know. We have this first case. In the second case, what I'll do is, I'll take this as positive and this as negative and we'll explore all the possibilities. Mathematics is all about exploring the possibilities. So, let's explore this one first. So, hence, if this is the case, so hence, I'm considering it case one, then you will get 5k is less than 8 or k is less than 8 upon 5. Okay? And if you solve this, you will get k should be greater than minus 8 upon 5. So, hence, clearly, the value of k will be minus 8 by 5 less than k less than 8 by 5. Isn't it? That means k must be a value between minus 8 by 5 and plus 8 by 5. This is where, so any k, so for example, 0, 1, minus 1, all these values, 1.1, all these value which lie between minus 8 by 5 and, you know, so and 8 by 5, those values will be, you know, the values of k for which the given equation will not have real roots, right? So, k could be anything between minus 8 by 5 and 8 by 5. So, infinitely, many possible values of k are there. Now, what if case two, the other way around, case two is 5k minus 8 is greater than 0 and 5k plus 8 is less than 0. This could be another possibility. So, from here, you'll get k is greater than 8 by 5 and k is less than minus 8 by 5. Now, guys, there exists no such k, no such k where both of them will be together. It's like saying a number which is greater than 2 as well as less than 2 or greater than 2, let's say, number greater than 2 and a number greater than minus 2. Can there be a number which is both greater than 2 as well as minus and less than minus 2? Not possible. Similarly, there will not be any number which is greater than 8 by 5 and simultaneously, less than minus 8 by 5. So, this is an impossible case. So, not possible. So, I can write this is not possible. So, hence, only one case is possible and that is case number one. And in that case, we are getting this particular inequality. So, any value of k between minus 8 by 5 and 8 by 5 will satisfy the condition that this given equation is having no real roots. Is that fine? So, that's the solution to the problem.