The value of m for the complex number to be purely real are 3 and -5.
The value of m for the complex number to be purely imaginary are -2 and 3.
For the complex number to be located in the second quadrant, the value of m must be less than -3 and 5.
Given the complex number:
[tex]z=\frac{m^2-m-6}{m+3}+(m^2-2m-15)i[/tex]
a) For the complex number to be purely real, then the imaginary part of the complex number must be zero that is:
[tex](m^2-2m-15)i = 0\\m^2-2m-15=0[/tex]
Factorize
[tex]m^2+5m-3m-15=0\\m(m+5)-3(m+5)=0\\(m-3)(m+5)=0\\m-3=0 \ and \ m+5=0\\m=3 \ and \ m=-5[/tex]
Hence the value of m for the complex number to be purely real are 3 and -5.
b) For the complex number to be purely imaginary, then the real part of the complex number must be zero. Hence;
[tex]\frac{m^2-m-6}{m+3}=0 \\m^2-m-6=0[/tex]
Factorize
[tex]m^2-m-6\\m^2-3m+2m-6=0\\m(m-3)+2(m-3)=0\\(m+2)(m-3)=0\\m+2=0 \ and \ m-3=0\\m=-2 \ and \ m = 3[/tex]
Hence the value of m for the complex number to be purely imaginary are -2 and 3.
c) For the complex number to be in the second quadrant, then the ratio of y to x must be negative i.e less than zero as shown:
[tex]\frac{m^2-2m-15}{\frac{m^2-m-6}{m+3} } < 0\\ \frac{m^2-2m-15(m+3)}{{m^2-m-6} }\\ \frac{(m+3)(m-5)(m+3)}{{(m-3)(m+2)} } <0\\(m+3)(m-5)(m+3) <0\\m+3<0, m-5<0 \ and \ m+3<0\\m<-3 \ and \ m<5[/tex]
Hence for the complex number to be located in the second quadrant, the value of m must be less than -3 and 5.
