Question

Find the quadratic equation whose one of the roots is \[2 + \sqrt 3 \].

Answer 1

Hint:- We have to assume that the required quadratic equation is of x. And then we equate x with the given root and on solving this equation by squaring both sides we will get the required quadratic equation.

Complete step-by-step answer:
Let the quadratic equation be of x.
So, as we know that one of the roots of the quadratic equation is \[2 + \sqrt 3 \].
\[ \Rightarrow \]So, x will also be equal to \[2 + \sqrt 3 \]
\[ \Rightarrow \]So, \[x = 2 + \sqrt 3 \] (1)
Now we can solve equation 1 to find the quadratic equation.
So, subtracting 2 from both the sides of the equation 1.
\[ \Rightarrow \]\[x - 2 = \sqrt 3 \] (2)
Squaring both sides of the above equation.
\[ \Rightarrow \]\[{\left( {x - 2} \right)^2} = {\left( {\sqrt 3 } \right)^2}\]
\[ \Rightarrow \]\[{x^2} - 4x + 4 = 3\]
\[ \Rightarrow \]\[{x^2} - 4x + 1 = 0\]
Hence, the quadratic equation whose one root is \[2 + \sqrt 3 \] will be \[{x^2} - 4x + 1 = 0\].

Note:- Whenever we come up with this type of problem then there is also an alternate method to find the quadratic equation. Like if one of the root of the quadratic equation is the sum of rational and irrational number (here \[2 + \sqrt 3 \]), then the other root must be the difference of same rational and irrational number (i.e. \[2 - \sqrt 3 \]). So, to find the quadratic equation we had to subtract both roots from x (i.e. \[\left( {x - \left( {2 + \sqrt 3 } \right)} \right)\] and \[\left( {x - \left( {2 - \sqrt 3 } \right)} \right)\]) and then multiply them to get the required quadratic equation (i.e. \[\left( {x - \left( {2 + \sqrt 3 } \right)} \right)\left( {x - \left( {2 - \sqrt 3 } \right)} \right) = {x^2} - 4x + 1\]).