Question

Factorise $x^2+2x-1$ by adding and subtracting appropriate quantity:

Answer 1

In the given equation $x^2+2x-1$, add and subtract $1^2$ to make it a perfect square as shown below:

$x^2+2x-1=(x^2+2x+1^2)-1-1^2=(x^2+(2\left)\right(1)x+1^2)-1-1=(x+1{)}^2-2$

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

Using the above identity, the expression $(x+1{)}^2-2$can be factorised as follows:

$(x+1{)}^2-2=(x+1{)}^2-(\sqrt{2}{)}^2=(x+1-\sqrt{2}\left)\right(x+1+\sqrt{2})$

Hence, $x^2+2x-1=(x+1-\sqrt{2})(x+1+\sqrt{2})$