Solve each equation for the specified variable. (Assume no denominators are 0.) See Example 8. $s=\left(\frac{1}{2}\right)g{t}^2$, for t

Verified step by step guidance

1. Start with the given equation: \(s = \frac{1}{2} g t^{2}\).
2. Multiply both sides of the equation by 2 to eliminate the fraction: \(2s = g t^{2}\).
3. Divide both sides by \(g\) to isolate \(t^{2}\): \(\frac{2s}{g} = t^{2}\).
4. Take the square root of both sides to solve for \(t\): \(t = \pm \sqrt{\frac{2s}{g}}\).
5. Remember to consider both the positive and negative roots since squaring either will give \(t^{2}\).

Key Concepts

Solving Equations for a Specific Variable

This involves isolating the desired variable on one side of the equation using algebraic operations such as addition, subtraction, multiplication, division, and taking roots. The goal is to rewrite the equation so that the specified variable is expressed explicitly in terms of the other variables.

Handling Quadratic Equations

When the variable to solve for appears squared, as in t², the equation is quadratic in that variable. Solving requires taking the square root of both sides, remembering to consider both positive and negative roots, unless context restricts the solution.

Avoiding Division by Zero

When rearranging equations, it is important to ensure that denominators are not zero, as division by zero is undefined. This often involves stating domain restrictions or assumptions explicitly to avoid invalid solutions.