Answer: The required value of m is
[tex]m=\dfrac{5+\sqrt{63}i}{4},~~~~~\dfrac{5-\sqrt{63}i}{4}.[/tex]
Step-by-step explanation: We are given to find the vale of m if :
[tex]m^2-\dfrac{5}{2}m=-\dfrac{11}{2}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
The given equation is a quadratic one, so we will be using the formula for the solution of a quadratic equation [tex]ax^2+bx+c=0,~a\neq 0[/tex] as follows :
[tex]x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}.[/tex]
From equation (i), we have
[tex]m^2-\dfrac{5}{2}m=-\dfrac{11}{2}\\\\\\\Rightarrow 2m^2-5m+11=0\\\\\Rightarrow x=\dfrac{-(-5)\pm\sqrt{(-5)^2-4\times2\times11}}{2\times2}\\\\\\\Rightarrow x=\dfrac{5\pm\sqrt{25-88}}{4}\\\\\\\Rightarrow x=\dfrac{5\pm\sqrt{-63}}{4}\\\\\\\Rightarrow x=\dfrac{5\pm\sqrt{63}i}{4}.[/tex]
Thus, the required value of m is
[tex]m=\dfrac{5+\sqrt{63}i}{4},~~~~~\dfrac{5-\sqrt{63}i}{4}.[/tex]
