In Exercises 45–47, solve each formula for the specified variable. vt + gt^2 = s for g

Verified step by step guidance

1. Start with the given equation: $vt\; +\; gt^2\; =\; s$. The goal is to solve for $g$.
2. Isolate the term containing $g$ by subtracting $vt$ from both sides: $gt^2\; =\; s\; -\; vt$.
3. Divide both sides of the equation by $t^2$ to solve for $g$: $g\; =\; \backslash (\backslash frac\{s\; -\; vt\}\{t^2\}\backslash )$.
4. Verify that $g$ is now isolated and expressed in terms of the other variables: $s$, $v$, and $t$.
5. The formula for $g$ is now fully solved: $g\; =\; \backslash (\backslash frac\{s\; -\; vt\}\{t^2\}\backslash )$. Ensure all variables are properly defined and understood in the context of the problem.

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate a specific variable. This process includes operations such as addition, subtraction, multiplication, and division applied to both sides of the equation. Understanding how to manipulate equations is crucial for solving for a variable, as it allows one to express the variable in terms of others.

Quadratic Equations

The equation vt + gt^2 = s is a quadratic equation in terms of g, where g is the variable to be solved for. Quadratic equations are polynomial equations of degree two and can often be rearranged into the standard form ax^2 + bx + c = 0. Recognizing the structure of quadratic equations is essential for applying methods such as factoring, completing the square, or using the quadratic formula.

Isolating Variables

Isolating a variable means rearranging an equation so that the variable appears on one side by itself. This often involves moving other terms to the opposite side of the equation and simplifying. In the context of the given equation, isolating g requires careful manipulation to ensure that all terms involving g are on one side, allowing for a clear solution.