Question

Factorise $6x^2+6x+3$ by adding and subtracting appropriate quantity:

Answer 1

In the given equation $x^2+6x+3$, add and subtract $3^2$ to make it a perfect square as shown below:

$x^2+6x+3=(x^2+6x+3^2)+3-3^2=(x^2+(2\left)\right(3)x+3^2)+3-9=(x+3{)}^2-6$

Since we know the identity $a^2-b^2=(a+b)(a-b)$

Using the above identity, the equation $(x+3{)}^2-6$can be factorised as follows:

$(x+3{)}^2-6=(x+3{)}^2-(\sqrt{6}{)}^2=(x+3-\sqrt{6}\left)\right(x+3+\sqrt{6})$

Hence, $x^2+6x+3=(x+3-\sqrt{6})(x+3+\sqrt{6})$