Problem 102 - College Algebra

Question

Solve for \(A: r-\sqrt{\frac{A}{4 \pi}}\)

Step-by-Step Solution

Answer

\( A = 4\pi r^2 \)

Step 1: Rewrite the equation

The given equation is \( r - \sqrt{\frac{A}{4\pi}} = 0 \). The main goal is to isolate \( A \) on one side. This can be achieved by moving terms with \( A \) on to one side of the equation and terms without \( A \) onto the other side. So, let's add \( \sqrt{\frac{A}{4\pi}} \) to both sides to get \( \sqrt{\frac{A}{4\pi}} = r \)

Step 2: Square both sides

To eliminate the square root, square both sides of the equation. This will give us \( \frac{A}{4\pi} = r^2 \)

Step 3: Isolate \( A \)

Now, to completely isolate \( A \), multiply both sides by \( 4\pi \). This will give us \( A = 4\pi r^2 \)

Key Concepts

Isolating Variables

Isolating variables is a fundamental technique in solving algebraic equations. It involves manipulating the equation so that the variable you are solving for is on one side of the equation and everything else is on the other side. Here's a simplified approach:

• Start by simplifying the equation as much as possible, combining like terms and removing any parentheses.
• Move all terms containing the variable to one side of the equation using addition or subtraction.
• Use multiplication or division to remove any coefficients or denominators associated with the variable.

In our example with the equation \( r - \sqrt{\frac{A}{4\pi}} = 0 \), we added \( \sqrt{\frac{A}{4\pi}} \) to both sides to move the term containing \(A\) to one side, effectively isolating the variable on one side of the equation.

Square Roots in Algebra

Dealing with square roots in algebra often requires the use of squaring to remove the radical. When you have an equation with a square root, you can square both sides of the equation to eliminate the square root. However, it's critical to remember that squaring both sides is an operation that can possibly introduce extraneous solutions, so you should always check your answers against the original equation.

In the step-by-step solution, we squared the isolated term \( \sqrt{\frac{A}{4\pi}} \) to eliminate the square root and obtain \( \frac{A}{4\pi} = r^2 \). By doing so, we transitioned from working with a radical equation to a more manageable algebraic equation.

Radical Equations

Radical equations are those that contain a variable within a radical, typically a square root. Solving these equations involves a few key steps:

• Isolate the radical expression involving the variable on one side of the equation.
• Square both sides of the equation to eliminate the square root.
• Solve the resulting quadratic equation if necessary.
• Check your solution by substituting it back into the original equation, because squaring can introduce false solutions.

For the initial problem, squaring both sides removed the radical and led us to an uncomplicated algebraic expression, allowing for the extraction of \( A \) in terms of \( r \). Always check the potential solutions in the original radical equation to verify their accuracy.