Question

The quadratic equation having the roots $(2+\sqrt{3})$ and $(2-\sqrt{3})$ is

• A. $x^2-4x+1=0$
• B. $x^2+4x-1=0$
• C. $x^2-4x-1=0$
• D. $x^2+4x+1=0$

Answer 1

The correct option is A $x^2-4x+1=0$

Given:

$(2+\sqrt{3})$ and $(2-\sqrt{3})$ are the roots of a quadratic equation.

$\therefore$ $x=(2+\sqrt{3})$ and $x=(2-\sqrt{3})$

$⟹$ $x-(2+\sqrt{3})=0$ and $x-(2-\sqrt{3})=0$

$⟹$ $(x-(2+\sqrt{3}\left)\right)(x-(2-\sqrt{3}\left)\right)=0$

Splitting two terms

$⟹$ $x^2-2x+\sqrt{3}x-2x-\sqrt{3}x+(2+\sqrt{3})(2-\sqrt{3})=0$

$⟹$ $x^2-4x+4-3=0$

$⟹$ $x^2-4x+1=0$

Hence, option A is correct.