Question

what is the solution to the equation (\frac{m}{m + 4}+\frac{4}{4 - m}=\frac{m^{2}}{m^{2}-16})?
options: (m = 2), (m = 4), (m=-2), (m = - 4)

Explanation:

Step1: Factor Denominators

Notice that \(4 - m=-(m - 4)\) and \(m^{2}-16=(m + 4)(m - 4)\) (difference of squares: \(a^{2}-b^{2}=(a + b)(a - b)\) with \(a = m\), \(b = 4\)). So the equation \(\frac{m}{m + 4}+\frac{4}{4 - m}=\frac{m^{2}}{m^{2}-16}\) can be rewritten as \(\frac{m}{m + 4}-\frac{4}{m - 4}=\frac{m^{2}}{(m + 4)(m - 4)}\).

Step2: Find Common Denominator

The common denominator for the left - hand side is \((m + 4)(m - 4)\). Multiply each fraction by the appropriate factor to get the common denominator:
\(\frac{m(m - 4)}{(m + 4)(m - 4)}-\frac{4(m + 4)}{(m + 4)(m - 4)}=\frac{m^{2}}{(m + 4)(m - 4)}\)

Step3: Simplify the Left - Hand Side

Expand the numerators:
\(m(m - 4)=m^{2}-4m\) and \(4(m + 4)=4m + 16\)
So the left - hand side becomes \(\frac{m^{2}-4m-(4m + 16)}{(m + 4)(m - 4)}=\frac{m^{2}-4m - 4m-16}{(m + 4)(m - 4)}=\frac{m^{2}-8m - 16}{(m + 4)(m - 4)}\)

Step4: Set Numerators Equal (Since Denominators are Equal and \(m

eq\pm4\) to avoid division by zero)
We have the equation \(m^{2}-8m - 16=m^{2}\) (since the denominators \((m + 4)(m - 4)\) are the same and non - zero for \(m
eq\pm4\), we can set the numerators equal).
Subtract \(m^{2}\) from both sides:
\(m^{2}-8m - 16-m^{2}=m^{2}-m^{2}\)
\(-8m-16 = 0\)
Add 16 to both sides: \(-8m=16\)
Divide both sides by \(-8\): \(m=- 2\)

We need to check if \(m = - 2\) makes the original denominators non - zero. For \(m=-2\), \(m + 4=-2 + 4 = 2
eq0\), \(4 - m=4-(-2)=6
eq0\), and \(m^{2}-16=(-2)^{2}-16=4 - 16=-12
eq0\). So \(m=-2\) is a valid solution.

Answer:

\(m=-2\) (the option with \(m = - 2\))