How can I factor $x^2 + 2\sqrt{3}\,x + 3$?

Question

$$x^2 + 2\sqrt{3}\,x + 3$$

Anyone could tell me how may I factor this? Thanks a lot

Answer 1

Since the $x^2$ does not have a number in front of it then you know it can be factorised into the form

$$ (x+A)(x+B)=x^2+(A+B)x+AB $$

So you want $A+B=2\sqrt{3}$ and $AB=3$. Can you see what $A$ and $B$ must be?

Answer 2

$$x^2+2\cdot \sqrt3 x+3$$

$$x^2+\sqrt 3x +\sqrt 3x+3$$

$$x(x+\sqrt 3) +\sqrt 3(x+\sqrt 3)$$

or, $$x=-\sqrt 3$$

Answer 3

You can factor this question in an number of ways. One way is to use the quadratic formula which you will give you roots and once you have the roots you can just rearrange and obtain the factored form. The quadratic formula is:

$$\frac{b \pm \sqrt{b^2-4ac}}{2a}$$

You have $b = 2\sqrt{3}$ and $c=3$ and $a = 1$. Sub those values in and you will get something like $x = a$ where $a$ is your answer and to get your factored form do $x - a$ and you're done!

Answer 4

Just know that: $$(x+a)^2=x^2+2ax+a^2.$$ And notice this: $$\begin{equation}\begin{split}x^2+2\sqrt{3}x+3&=x^2+2\sqrt{3}x+\left(\sqrt{3}\right)^2\\&=(x+\sqrt{3})^2.\end{split}\end{equation}$$