Problem 53 - Elementary Algebra

Question

Paola has enough mulch to cover 48 square feet. She wants to use it to make three square vegetable gardens of equal sizes. Solve the equation \(3 s^{2}=48\) to find \(s,\) the length of each garden side.

Step-by-Step Solution

Answer

The length of each garden side is 4 feet.

Step 1: Identify the equation

The given equation is \(3s^{2} = 48\). This equation represents the total area covered by the mulch for three square vegetable gardens.

Step 2: Isolate the variable term

To find the value of \(s\), first isolate the square term by dividing both sides of the equation by 3. This gives: \[ s^{2} = \frac{48}{3} \]}, {

Step 3: Simplify the equation

Simplify \(\frac{48}{3}\) to get 16. Now, the equation is: \( s^{2} = 16 \)

Step 4: Solve for the variable

To solve for \(s\), take the square root of both sides of the equation: \( s = \sqrt{16} \) which simplifies to \( s = 4 \).

Key Concepts

Isolating the Variable

To solve an equation, the first step often involves isolating the variable you're solving for. In this problem, we need to find the value of \(s\), the side length of each garden. Starting with the equation \(3s^{2} = 48\),
we need to get \(s\) by itself on one side of the equation.

This can be done by performing the same operation on both sides of the equation. Here, we divide both sides of the equation by 3 so that we can eliminate the coefficient of the \(s^2\) term. Doing this, we have:
\[ s^{2} = \frac{48}{3} \ s^{2} = 16 \]

Simplifying Equations

Simplifying equations is about making them easier to handle. The goal is to reduce the expression down to the simplest form.
In this step, we took the equation \( s^{2} = \frac{48}{3} \) and simplified the fraction. By dividing 48 by 3, we got 16, which leads us to the equation \( s^{2} = 16 \).

Simplifying the equation helps us to focus directly on solving for the variable \(s\).
Here are a few tips for simplifying equations effectively:

• Combine like terms.
• Reduce fractions by finding the greatest common divisor.
• Cancel out terms that appear on both sides of the equation if applicable.
• Factor expressions whenever possible.

Square Roots

Taking square roots is a common method to solve quadratic equations like the one we have here:
\( s^{2} = 16 \).

To find the value of \(s\), we take the square root of both sides of the equation:
\[ s = \text{sqrt}(16) = 4 \ s = \text{sqrt}(16) = -4 \]
However, in the context of this problem, we are considering the size of the vegetable garden, so we use the positive value:
\( s = 4 \).

It's important to remember that every non-zero number has two square roots (one positive and one negative).

Here are some tips to consider when solving for roots:

• Identify if the problem context requires only the positive root.
• Always check your work by squaring your root to see if it returns the original number.
• For more complex numbers, use a calculator or software to ensure accuracy.