Question

what is the value of k if one of the root of quadratic equation X² - 4x +k = 0, is 2+ root 3​

Answer 1

Answer:

The value of k is 1.

Step-by-step-explanation:

We have given that,

One root of the quadratic equation x² - 4x + k = 0 is $\displaystyle{\sf\:2\:+\:\sqrt{3}}$

We have to find the value of k.

The given quadratic equation is

$\displaystyle{\sf\:x^2\:-\:4x\:+\:k\:=\:0}$

$\displaystyle{\implies\sf\:k\:=\:4x\:-\:x^2}$

By substituting $\displaystyle{\sf\:x\:=\:2\:+\:\sqrt{3}}$, we get,

$\displaystyle{\implies\sf\:k\:=\:4\:(\:2\:+\:\sqrt{3}\:)\:-\:(\:2\:+\:\sqrt{3}\:)^2}$

$\displaystyle{\implies\sf\:k\:=\:8\:+\:4\:\sqrt{3}\:-\:[\:(\:2\:)^2\:+\:2\:\times\:2\:\times\:\sqrt{3}\:+\:(\:\sqrt{3}\:)^2\:]}$

$\displaystyle{\implies\sf\:k\:=\:8\:+\:4\:\sqrt{3}\:-\:(\:4\:+\:4\:\sqrt{3}\:+\:3\:)}$

$\displaystyle{\implies\sf\:k\:=\:8\:+\:\cancel{4\:\sqrt{3}}\:-\:4\:-\:\cancel{4\:\sqrt{3}}\:-\:3}$

$\displaystyle{\implies\sf\:k\:=\:8\:-\:4\:-\:3}$

$\displaystyle{\implies\sf\:k\:=\:4\:-\:3}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:k\:=\:1\:}}}}$

∴ The value of k is 1.