Question

If one root of the equation $x^2+px+q=0$ is $2+\sqrt{3},$ then the values of $p$ and $q$ are respectively.

• A. $-4,1$
• B. $4,-1$
• C. $2,\sqrt{3}$
• D. $-2-\sqrt{3}$

Answer 1

The correct option is A $-4,1$

$\Rightarrow$ The given quadratic equation is $x^2+px+q=0$ --- ( 1 )

$\Rightarrow$ One root of equation ( 1 ) is $2+\sqrt{3}$

$\Rightarrow$ If $p$ and $q$ are rational, then unique value of $p$ and $q$ exist.

$\Rightarrow$ If they are not rational, $p$ $q$ may have infinite value as the other root may or may not be irrational.

Now, let us assume that $p$ and $q$ are rational numbers.

Then, other root of equation ( 1 ) must be $2-\sqrt{3}$.

$\Rightarrow$ Sum of the roots $=-p$

$\Rightarrow$ $-p=2+\sqrt{3}+2-\sqrt{3}$

$\Rightarrow$ $-p=4$

$\therefore$ $p=-4$

Product of roots $=q$

$\Rightarrow$ $q=(2+\sqrt{3})(2-\sqrt{3})$

$\Rightarrow$ $q=4-3$

$\Rightarrow$ $q=1$

$\therefore$ $p=-4$ and $q=1$