Question

solve for ( s ). if there are multiple solutions, enter them as a list of values separated by commas. if there are no solutions enter none.
\frac{s + 4}{s^2 + 14s + 48} = -\frac{3}{s + 6} + \frac{2}{s + 8}
( s = )
thats not it. try again.

Explanation:

Step1: Factor the denominator

Factor \(s^2 + 14s + 48\) as \((s + 6)(s + 8)\). So the equation becomes \(\frac{s + 4}{(s + 6)(s + 8)} = -\frac{3}{s + 6} + \frac{2}{s + 8}\).

Step2: Multiply through by \((s + 6)(s + 8)\)

Multiply each term by \((s + 6)(s + 8)\) to eliminate denominators: \(s + 4 = -3(s + 8) + 2(s + 6)\).

Step3: Expand and simplify

Expand the right side: \(s + 4 = -3s - 24 + 2s + 12\). Combine like terms: \(s + 4 = -s - 12\).

Step4: Solve for \(s\)

Add \(s\) to both sides: \(2s + 4 = -12\). Subtract 4: \(2s = -16\). Divide by 2: \(s = -8\).

Step5: Check for extraneous solutions

Check if \(s = -8\) makes any original denominator zero. The denominators are \((s + 6)(s + 8)\), \(s + 6\), and \(s + 8\). When \(s = -8\), \(s + 8 = 0\), so \(s = -8\) is extraneous. Thus, no solution.

Answer:

None