 Hello and welcome to the session. In this session we discussed a following question which says check whether the roots of the equation x-b whole multiplied by x-c plus x-a whole multiplied by x-b plus x-c whole multiplied by x-a equal to 0 are real or not. Also find the condition for the roots to be equal. For the given equation we have to check if the roots of this equation are real or not and we also have to find the condition such that the roots of the given equation are equal. So first let's check out the condition when the roots of the given quadratic equation are real. Consider the quadratic equation ax square plus bx plus c equal to 0 where we have a is not equal to 0 then the discriminant d is given by b square minus 4ac. The roots of the quadratic equation are real if the discriminant d is greater than equal to 0 and the roots of the quadratic equation are equal if we have discriminant d is equal to 0. This is the key idea that we use in this question. Now we proceed with its solution. The given quadratic equation is x minus b whole into x minus c plus x minus a whole into x minus b plus x minus c whole into x minus a is equal to 0. So this equation can also be written as x square minus b plus c whole into x plus bc plus x square minus a plus b whole into x plus ab plus x square minus a plus c whole into x plus ac is equal to 0. This can be rewritten as 3x square minus b plus c plus a plus b plus a plus c this whole into x plus bc plus ab plus ac equal to 0 or we can also say 3x square minus 2x into a plus b plus c plus ab plus bc plus ca is equal to 0. Now we need to find the discriminant d and it is given by b square minus 4ac for this equation b is equal to the coefficient of x which is minus 2 into a plus b plus c, a is equal to the coefficient of x square which is 3 and c is equal to the constant which is ab plus bc plus ca. So discriminant d is equal to we will put the values for ab and c here and this would give us d is equal to minus 2 into a plus b plus c whole square minus 4 into 3 into ab plus bc plus ca. So we have d is equal to 4 into a plus b plus c whole square minus 12 into ab plus bc plus ca. This further gives us d is equal to 4 into a plus b plus c whole square can be written as a square plus b square plus c square plus 2ab plus 2bc plus 2ca minus 12 into ab plus bc plus ca. That is d is equal to 4 into a square plus b square plus c square minus ab minus bc minus ca. This can also be written as d is equal to 2 into 2a square plus 2b square plus 2c square minus 2ab minus 2bc minus 2ca. So we now get d is equal to 2 into a square plus b square minus 2ab plus a square plus c square minus 2ac or 2ca plus b square plus c square minus 2bc which further gives d is equal to 2 into a minus b whole square plus a minus c whole square plus b minus c whole square. Now since we know that a minus b whole square is greater than equal to 0 a minus c whole square is greater than equal to 0 and b minus c whole square is greater than equal to 0 therefore this d is greater than equal to 0. Now that we have got d is greater than equal to 0 so this shows that both the roots of the given equation real. Next we have to find the condition for the roots to be equal. So for equal roots we have discriminant d is equal to 0. Now when d is equal to 0 this means 2 into a minus b whole square plus a minus c whole square plus b minus c whole square is equal to 0 or you can say a minus b whole square plus a minus c whole square plus b minus c whole square is equal to 0. So this means we have a minus B is equal to 0, A minus C is equal to 0 and also B minus C is equal to 0 that is A is equal to B, A is equal to C and B is equal to C. Therefore from these three conditions we get one condition which is A is equal to B is equal to C. So we say the roots of the given quadratic equation are equal only when A is equal to B is equal to C. So this is the required condition. This completes the session. Hope you have understood the solution for this question.