Here in 6x^2-36x2−3, there is no middle term and polynomial appears as difference of remaining two terms. In such cases, we can write it as a^2-b^2a2−b2 and factors are (a+b)(a-b)(a+b)(a−b), while zeros can be obtained by putting a+b=0a+b=0 and a-b=0a−b=0.
Here, however, one desires to use the method of splitting the middle term and for this one can add and subtract abab. Here as a^2=6x^2a2=6x2, our a=xsqrt6a=x√6 and as 3=(sqrt3)^23=(√3)2, we add and subtract xsqrt6xxsqrt3=sqrt18xx√6×√3=√18x and then
6x^2-36x2−3
= 6x^2+sqrt18x-sqrt18x-36x2+√18x−√18x−3
= sqrt6x(sqrt6x+sqrt3)-sqrt3(sqrt6x+sqrt3)√6x(√6x+√3)−√3(√6x+√3)
= (sqrt6x-sqrt3)(sqrt6x+sqrt3)(√6x−√3)(√6x+√3)
Here as middle term is 00, we have split it into +sqrt18x-sqrt18x+√18x−√18x
and zeros are given by sqrt6x-sqrt3=0√6x−√3=0 i.e. x=sqrt3/sqrt6=1/sqrt2x=√3√6=1√2
and sqrt6x+sqrt3=0√6x+√3=0 i.e. x=-sqrt3/sqrt6=-1/sqrt2x=−√3√6=−1√2
