Answer :

Sure, let's solve the given equation [tex]\( y = \frac{\sqrt{t^2} - x^2}{v^3} \)[/tex] for [tex]\( x \)[/tex]:

1. Start with the given equation:
[tex]\[ y = \frac{\sqrt{t^2} - x^2}{v^3} \][/tex]

2. Isolate the fraction by multiplying both sides by [tex]\( v^3 \)[/tex]:
[tex]\[ y \cdot v^3 = \sqrt{t^2} - x^2 \][/tex]

3. Simplify [tex]\( \sqrt{t^2} \)[/tex]:
[tex]\[ \sqrt{t^2} = |t| \][/tex]
Therefore, the equation becomes:
[tex]\[ y \cdot v^3 = |t| - x^2 \][/tex]

4. Isolate the term involving [tex]\( x^2 \)[/tex] by subtracting [tex]\( |t| \)[/tex] from both sides:
[tex]\[ y \cdot v^3 - |t| = -x^2 \][/tex]

5. Change the sides of the equation for clarity:
[tex]\[ -x^2 = y \cdot v^3 - |t| \][/tex]
Multiply both sides by -1:
[tex]\[ x^2 = - (y \cdot v^3 - |t|) \][/tex]
Simplify the right-hand side:
[tex]\[ x^2 = -y \cdot v^3 + |t| \][/tex]

6. Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \pm\sqrt{-y \cdot v^3 + |t|} \][/tex]

Thus, the solutions for [tex]\( x \)[/tex] are:
[tex]\[ x = \sqrt{-y \cdot v^3 + |t|} \quad \text{and} \quad x = -\sqrt{-y \cdot v^3 + |t|} \][/tex]