Question

solve for t. if there are multiple solutions, enter them as a list of values separated by commas. if there are no solutions, type none.
\\(\frac{2}{t - 8} + \frac{4}{t + 8} = \frac{3}{t^2 - 64}\\)

t =

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.

t =

Explanation:

Step1: Factor the denominator

Notice that \( t^2 - 64=(t - 8)(t + 8) \), so the equation is \(\frac{2}{t - 8}+\frac{4}{t + 8}=\frac{3}{(t - 8)(t + 8)}\).

Step2: Multiply through by \((t - 8)(t + 8)\)

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

Step3: Expand the brackets

Expand the left - hand side: \(2t+16 + 4t-32 = 3\).

Step4: Combine like terms

Combine the \(t\) terms and the constant terms: \(6t-16 = 3\).

Step5: Solve for \(t\)

Add 16 to both sides: \(6t=3 + 16=19\), then divide by 6: \(t=\frac{19}{6}\).

Step6: Check for extraneous solutions

We need to check if \(t=\frac{19}{6}\) makes the original denominators zero. The denominators are \(t - 8=\frac{19}{6}-8=\frac{19 - 48}{6}=-\frac{29}{6}
eq0\) and \(t + 8=\frac{19}{6}+8=\frac{19 + 48}{6}=\frac{67}{6}
eq0\), so there are no extraneous solutions.

Answer:

(for \(t\)):
\(\frac{19}{6}\)