13 It is given that $$ t=\frac{b^{2}-4}{3-b^{2}} $$ Express $$ b $$ in terms of $$ t $$

Question

UpStudy AI · Accepted Answer

We start with the equation
$$
t = \frac{b^{2} - 4}{3 - b^{2}}
$$
and solve for $$ b $$.

1. Multiply both sides by $$(3 - b^{2})$$ (assuming $$b^2 \neq 3$$):
   $$
   t(3 - b^{2}) = b^{2} - 4
   $$

2. Distribute $$t$$ on the left-hand side:
   $$
   3t - t b^{2} = b^{2} - 4
   $$

3. Get all terms involving $$b^{2}$$ on one side. Add $$t b^{2}$$ to both sides and also add 4 to both sides:
   $$
   3t + 4 = b^{2} + t b^{2}
   $$

4. Factor $$b^{2}$$ on the right-hand side:
   $$
   3t + 4 = b^{2}(1 + t)
   $$

5. Solve for $$b^{2}$$ assuming $$1 + t \neq 0$$ (i.e., $$t \neq -1$$):
   $$
   b^{2} = \frac{3t + 4}{1 + t}
   $$

6. Finally, take the square root of both sides:
   $$
   b = \pm \sqrt{\frac{3t + 4}{1 + t}}
   $$

Thus, the solution expressing $$ b $$ in terms of $$ t $$ is:
$$
b = \pm \sqrt{\frac{3t + 4}{1 + t}}
$$
$$ b = \pm \sqrt{\frac{3t + 4}{1 + t}} $$

13 It is given that \( t=\frac{b^{2}-4}{3-b^{2}} \) Express \( b \) in terms of \( t \)