Solve each equation for the specified variable. (Assume no denominators are 0.) See Example 8.
$r=r_0+\left(\frac{1}{2}\right)a{t}^2$, for t

Verified step by step guidance

1. Start with the given equation: \(r = r_0 + \frac{1}{2} a t^2\).
2. Isolate the term containing \(t^2\) by subtracting \(r_0\) from both sides: \(r - r_0 = \frac{1}{2} a t^2\).
3. Eliminate the fraction by multiplying both sides of the equation by 2: \$2 (r - r_0) = a t^2$.
4. Solve for \(t^2\) by dividing both sides by \(a\): \(t^2 = \frac{2 (r - r_0)}{a}\).
5. Take the square root of both sides to solve for \(t\): \(t = \pm \sqrt{\frac{2 (r - r_0)}{a}}\).

Key Concepts

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 factoring. The goal is to rewrite the equation so that the specified variable is expressed explicitly in terms of the other variables.

Handling Fractions in Algebraic Equations

When an equation contains fractions, it is important to clear denominators by multiplying both sides by the least common denominator or carefully apply inverse operations. This simplifies the equation and helps isolate the variable without introducing extraneous solutions.

Quadratic Expressions and Square Roots

Equations involving squared variables, like t², require taking square roots to solve for the variable. Remember to consider both positive and negative roots when solving, and ensure the solution fits the context of the problem.