13 It is given that \( t=\frac{b^{2}-4}{3-b^{2}} \) Express \( b \) in terms of \( t \)
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To express \( b \) in terms of \( t \), we start with the equation \( t = \frac{b^{2} - 4}{3 - b^{2}} \). First, we can cross-multiply to eliminate the fraction: \[ t(3 - b^{2}) = b^{2} - 4. \] Expanding this gives: \[ 3t - tb^{2} = b^{2} - 4. \] Next, let's rearrange the equation to get all terms involving \( b^{2} \) on one side: \[ tb^{2} + b^{2} = 3t + 4. \] This can be factored into: \[ b^{2}(t + 1) = 3t + 4. \] Now, divide both sides by \( t + 1 \) (assuming \( t + 1 \neq 0 \)): \[ b^{2} = \frac{3t + 4}{t + 1}. \] Finally, taking the square root gives us: \[ b = \pm \sqrt{\frac{3t + 4}{t + 1}}. \] Thus, \( b \) can be expressed in terms of \( t \) as: \[ b = \pm \sqrt{\frac{3t + 4}{t + 1}}. \]
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