 Ok, welcome again to this session on quadratic equation folks. Now in this session we are going to take up how to find out the condition that the roots of a quadratic equation A x square plus B x plus c equals 0 is A equal in magnitude, either the first condition is an opposite sign and secondly receive locals. Ok, so we have to find conditions such that the roots of a given quadratic equation are equal in magnitude and opposite sign and second is what is the condition for them to be receive locals of each other. So, let us take up case A right. So, roots are equal and opposite in sign opposite in sign. So, if you see we have learned that you know the properties of the roots and its relation and the relation with the coefficients. So, we will use that in this case. So, let the roots be let the roots be alpha and minus alpha. Ok, now sum of roots sum of roots of equation A x square plus B x plus c equals 0 is given by minus B upon A you have seen that in the previous sessions. So, minus B by A. So now, so root sum of roots will be alpha plus minus alpha is equal to minus B upon A. This implies minus B upon A is 0 0 right minus B upon A is 0. So, that means B has to be 0 B has to be 0 right. So, this is the condition. So, examples. So, let us take examples. So, hence examples like x square minus 4 equals 0. In this case, if you see x is equal to plus minus 2 is it not? So, hence x equals to plus minus 2. So, equal and opposite signs is it not? And similarly 2 x square is equal to let us say 3. So, x square will be equal to 3 by 2. So, x will be equal to plus minus under root 3 by 2. So, equal in magnitude and opposite sign. So, there will not be any term containing x. The moment you see such equations you know that roots are equal and opposite. What about case B? Reciprocal roots. Reciprocal roots. Reciprocal roots right. So, let us say roots are let us say roots are alpha and 1 upon alpha. Okay. These are the 2 roots and we know that product of roots. So, you will use this condition product of roots product of roots of equation A x square plus B x plus C equals 0 is nothing but C upon A is it not? So, hence we get alpha times 1 upon alpha is C upon A and this means C by A equals 1 that means A is equal to C. So, this is the condition for reciprocal roots guys. Example. So, that means the constant term must be equal to the coefficient of x square. So, hence x square minus 2x plus 1 is equal to 0. This will have reciprocal roots right. Similarly, x square minus 4x plus 2 is equal to 0. This will also have no this will not have reciprocal roots. Why? Because C is not equal to yeah. So, plus 1 this will have reciprocal roots. Isn't it? And similarly, let us say 2x square 2x square minus 5x plus 2 equals to 0. So, this will also have reciprocal roots. Let us solve this and C. A is equal to C here. Can you check? Yes. A is equal to C. So, hence by this rule the root should be reciprocal right. So, hence let us try to solve this. So, 2x square minus 4x minus x plus 2 equals 0. So, 2x common and hence x minus 2 and here minus x minus 2 equals 0. So, hence you will get x minus 2 times 2x minus 1 equals 0. So, you clearly see either x equals to 2 or x equals to 1 upon 2. So, you can see they are reciprocal to each other. So, hence it works. So, if A is equal to C we have reciprocal roots. This is the condition. Okay. So, this will be very effective knowledge for solving questions quickly in a comparative exam.