Solve the following equation for \( t \) : \[ \frac{8}{t}-\frac{14}{t+6}=\frac{168}{t^{2}-36} \] Help: If there is more than one correct answer, enter our answers as a comma separated list. If the equation tas no real solution, write "No Solution."

To solve the equation \[ \frac{8}{t} - \frac{14}{t+6} = \frac{168}{t^{2}-36}, \] first, note that \( t^2 - 36 = (t-6)(t+6) \). We can rewrite the equation as: \[ \frac{8}{t} - \frac{14}{t+6} = \frac{168}{(t-6)(t+6)}. \] Next, let's find a common denominator for the left side. The common denominator will be \( t(t+6) \). Rewriting both fractions yields: \[ \frac{8(t+6) - 14t}{t(t+6)} = \frac{168}{(t-6)(t+6)}. \] Expanding the left side: \[ \frac{8t + 48 - 14t}{t(t+6)} = \frac{168}{(t-6)(t+6)}. \] This simplifies to: \[ \frac{-6t + 48}{t(t+6)} = \frac{168}{(t-6)(t+6)}. \] Cross-multiplying gives: \[ (-6t + 48)(t - 6) = 168t(t + 6). \] Expanding both sides: \[ -6t^2 + 36t + 48t - 288 = 168t^2 + 1008t. \] \[ -6t^2 + 84t - 288 = 168t^2 + 1008t. \] Combining and rearranging: \[ 0 = 168t^2 + 1008t + 6t^2 - 84t + 288, \] \[ 0 = 174t^2 + 924t + 288. \] Now, we can factor this or use the quadratic formula. We will use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): Here, \( a = 174 \), \( b = 924 \), \( c = 288 \): Calculating the discriminant: \[ b^2 - 4ac = 924^2 - 4(174)(288). \] \[ = 853776 - 200256 = 653520. \] Now, since \( 653520 \) is a positive number, we can find the roots: \[ t = \frac{-924 \pm \sqrt{653520}}{2 \cdot 174}. \] Calculating the square root: \[ \sqrt{653520} \approx 808.83, \] thus, \[ t = \frac{-924 \pm 808.83}{348}. \] Calculating both possible solutions: 1. \( t = \frac{-924 + 808.83}{348} \approx -0.3300 \) 2. \( t = \frac{-924 - 808.83}{348} \approx -4.2710 \) Thus, the solutions to the equation are: \[ \boxed{-0.33, -4.27}. \]