Question

solve for m. if there are multiple solutions, enter them as a list of values separated by commas. if there are no solutions, enter none.\\(\frac{m}{m + 8} = \frac{128}{m^2 - 64} + 9\\)\\(m = \\)\\(\\)enter any extraneous solutions. if there are multiple extraneous solutions, enter them as a list separated by a comma. if there are no extraneous solutions, enter none.\\(m = \\)\\(\\)

Explanation:

Step1: Factor the denominator

Notice that \(m^2 - 64=(m + 8)(m - 8)\) by the difference of squares formula \(a^2 - b^2=(a + b)(a - b)\). So the equation \(\frac{m}{m + 8}=\frac{128}{m^2 - 64}+9\) can be rewritten as \(\frac{m}{m + 8}=\frac{128}{(m + 8)(m - 8)}+9\). The domain of the original equation requires \(m
eq - 8\) and \(m
eq8\) (since these values would make the denominators zero).

Step2: Multiply through by the common denominator

Multiply each term by \((m + 8)(m - 8)\) to clear the fractions:
\(m(m - 8)=128+9(m + 8)(m - 8)\)

Step3: Expand the expressions

Expand the left - hand side: \(m^2-8m = 128+9(m^2 - 64)\)
Expand the right - hand side: \(m^2-8m=128 + 9m^2-576\)

Step4: Simplify the equation

First, simplify the right - hand side: \(m^2-8m=9m^2 - 448\)
Then, move all terms to one side: \(m^2-8m-9m^2 + 448 = 0\)
Combine like terms: \(-8m^2-8m + 448 = 0\)
Divide through by \(-8\) to simplify: \(m^2+m - 56=0\)

Step5: Solve the quadratic equation

Factor the quadratic equation \(m^2+m - 56 = 0\). We need two numbers that multiply to \(-56\) and add up to \(1\). The numbers are \(8\) and \(-7\). So, \(m^2+m - 56=(m + 8)(m - 7)=0\)
Set each factor equal to zero: \(m+8 = 0\) or \(m - 7=0\)
We get \(m=-8\) or \(m = 7\)

Step6: Check for extraneous solutions

Recall from the domain that \(m
eq - 8\) (because \(m=-8\) makes the original denominator \(m + 8 = 0\)). So \(m=-8\) is an extraneous solution. We check \(m = 7\):
Left - hand side: \(\frac{7}{7 + 8}=\frac{7}{15}\)
Right - hand side: \(\frac{128}{7^2-64}+9=\frac{128}{49 - 64}+9=\frac{128}{-15}+9=\frac{-128 + 135}{15}=\frac{7}{15}\)
So \(m = 7\) is a valid solution.

Answer:

(for the first part, solving for \(m\)):
\(7\)