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

Verified step by step guidance

1. Step 1: The given equation is $vt + gt^2 = s$. We are asked to solve for $g$.
2. Step 2: To isolate $g$, we first need to get rid of the $vt$ term on the left side of the equation. We can do this by subtracting $vt$ from both sides of the equation. This gives us $gt^2 = s - vt$.
3. Step 3: Now, we need to isolate $g$. We can do this by dividing both sides of the equation by $t^2$. This gives us $g = \frac{s - vt}{t^2}$.
4. Step 4: This is the final form of the equation, solved for $g$.
5. Step 5: Always remember to check your solution by substituting the value of $g$ back into the original equation to ensure it holds true.

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate a specific variable. This process includes operations such as adding, subtracting, multiplying, or dividing both sides of the equation by the same value. Understanding how to manipulate equations is essential for solving for a variable, especially in formulas that contain multiple terms.

Quadratic Equations

The equation vt + gt^2 = s is a quadratic equation in terms of g, where g is the variable we want to isolate. Quadratic equations are polynomial equations of degree two and can often be solved using methods such as factoring, completing the square, or applying the quadratic formula. Recognizing the structure of a quadratic equation is crucial for finding solutions.

Isolating Variables

Isolating a variable means rearranging an equation so that the variable appears on one side by itself. In this case, we need to express g in terms of the other variables (v, t, and s). This process often requires moving other terms to the opposite side of the equation and may involve dividing by coefficients, which is a fundamental skill in algebra.