Answer: The correct options are
(B) [tex]x=\dfrac{11+\sqrt{61}}{2}.[/tex]
(D) [tex]x=\dfrac{11-\sqrt{61}}{2}.[/tex]
Step-by-step explanation: We are given to select the values of x that are the roots of the following polynomial :
[tex]x^2-11x+15.[/tex]
The quadratic equation formed by the given polynomial will be
[tex]x^2-11x+15=0~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
we know that
the solution set of a quadratic equation [tex]ax^2+bx+c=0,~~a\neq 0[/tex] is given by
[tex]x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}.[/tex]
From equation (i), we have
a = 1, b = -11 and c = 15.
Therefore, the roots of equation (i) will be given by
[tex]x=\dfrac{-(-11)\pm\sqrt{(-11)^2-4\times1\times15}}{2\times1}\\\\\\\Rightarrow x=\dfrac{11\pm\sqrt{121-60}}{2}\\\\\\\Rightarrow x=\dfrac{11\pm\sqrt{61}}{2}.[/tex]
Thus, the roots of the given polynomial are
[tex]x=\dfrac{11+\sqrt{61}}{2},~~~~~x=\dfrac{11-\sqrt{61}}{2}.[/tex]
Options (B) and (D) are CORRECT.
