Answer:
The solution is m = -2
Step-by-step explanation:
The given rational equation is
[tex]\frac{m}{m+4}+\frac{4}{4-m}=\frac{m^2}{m^2-16}[/tex]
We'll simplify the left hand side first.
The LCD is (m+4)(4-m)
Hence, multiply and divide the first term by 4-m and second by m +4
[tex]\frac{m(4-m)}{(m+4)(4-m)}+\frac{4(m+4)}{(4-m)(m+4}=\frac{m^2}{m^2-16}[/tex]
Use the difference of squares rule [tex]a^2-b^2=(a+b)(a-b)[/tex]
[tex]\frac{m(4-m)}{4^2-m^2}+\frac{4(m+4)}{4^2-m^2}=\frac{m^2}{m^2-16}[/tex]
We can now add the numerator
[tex]\frac{m(4-m)+4(m+4)}{4^2-m^2}=\frac{m^2}{m^2-16}[/tex]
On simplifying, we get
[tex]\frac{4m-m^2+4m+16}{16-m^2}=\frac{m^2}{m^2-16}[/tex]
Add the like terms
[tex]\frac{8m-m^2+16}{16-m^2}=\frac{m^2}{m^2-16}[/tex]
Multiply and divide left hand side by -1
[tex]\frac{m^2-8m-16}{m^2-16}=\frac{m^2}{m^2-16}[/tex]
We can cancel the denominator
[tex]-8m-16=0\\8m=-16[/tex]
Divide both sides by 8
[tex]m=-2[/tex]
The solution is m = -2
