Solve each equation for the specified variable. (Assume no denominators are 0.) See Example 8. r = r_0+(1/2)at^2, for t

Verified step by step guidance

1. Start with the equation: $r=r_0+\frac{1}{2}a{t}^2$.
2. Subtract $r_0$ from both sides to isolate the term with $t$: $r-r_0=\frac{1}{2}a{t}^2$.
3. Multiply both sides by 2 to eliminate the fraction: $2(r-r_0)=a{t}^2$.
4. Divide both sides by $a$ to solve for $t^2$: $t^2=\frac{2(r-r_0)}{a}$.
5. Take the square root of both sides to solve for $t$: $t=\pm \sqrt{\frac{2(r-r_0)}{a}}$.

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate a specific variable. This 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, as it allows you to express one variable in terms of others.

Quadratic Equations

The equation given, r = r_0 + (1/2)at^2, is a form of a quadratic equation when rearranged. Quadratic equations are polynomial equations of degree two, typically in the form ax^2 + bx + c = 0. Recognizing the structure of quadratic equations is important for applying appropriate methods to solve for the variable, such as factoring or using the quadratic formula.

Isolating Variables

Isolating a variable means rearranging an equation so that the variable of interest is on one side of the equation by itself. This process often involves inverse operations to eliminate other terms. In the context of the given equation, isolating t requires moving other terms to the opposite side and applying square roots, which is a critical skill in algebra.