Problem 10 - Prealgebra

Question

Solve each equation. Round to the nearest tenth, if necessary. $$3 m^{2}=222$$

Step-by-Step Solution

Answer

The solutions for \(m\) are approximately \(m \approx 8.6\) and \(m \approx -8.6\).

Step 1: Isolate the variable term

Start by dividing both sides of the equation by 3 to isolate the term containing the variable. This gives us:\[m^2 = \frac{222}{3}\] Calculate \(\frac{222}{3}\) to get:\[m^2 = 74\]

Step 2: Solve for the variable

To find \(m\), we need to take the square root of both sides of the equation to solve for \(m\):\[m = \pm \sqrt{74}\]

Step 3: Approximate the solution

Calculate \(\sqrt{74}\) using a calculator to get approximately 8.6. Since we need \(m\) to the nearest tenth, the solutions are:\[m \approx \pm 8.6\]

Key Concepts

Isolating Variables

Isolating variables is an essential first step in solving algebraic equations, especially when dealing with quadratic equations. The goal here is to get the variable, in this case, \(m\), by itself on one side of the equation. This makes it easier to solve for the variable's value.

Here's how you do it:

• Look at the equation. Identify the term with the variable. In our example, it’s \(3m^2\).
• Move other terms away from the variable term, if necessary, by adding, subtracting, multiplying, or dividing both sides of the equation as needed.
• For the equation \(3m^2 = 222\), divide both sides by 3 to isolate \(m^2\):

This leaves you with \(m^2 = 74\). Now that the variable term is isolated, you can focus on solving the equation by addressing the squared term.

Taking Square Roots

Once the variable is isolated and squared, the next step to solving the equation is to take the square root. This is because you need to undo the squared term to find the value of the variable.

Here's the process:

• Identify the equation you need to solve, which now looks like \(m^2 = 74\).
• Take the square root of both sides of this equation. Remember, when you take the square root of both sides, you'll get two potential solutions, positive and negative.
• This means \(m = \pm \sqrt{74}\).

Taking square roots allows us to find the possible values that the variable \(m\) can take, effectively solving the quadratic equation.

Approximation of Square Roots

Sometimes, square roots do not yield whole number solutions, and you need to approximate them to solve the equation. Approximating square roots is common, especially in practical scenarios. You typically use a calculator to achieve this.

Here's how:

• Use a calculator to approximate \(\sqrt{74}\).
• Calculate as accurately as needed. Here, \(\sqrt{74} \approx 8.6\).
• Ensure you consider both the positive and negative values, thus giving \(m \approx \pm 8.6\).

Approximating square roots helps to provide a more practical numeric answer when dealing with square roots that do not simplify neatly into integers.

Rounding to the Nearest Tenth

In mathematics, precision is crucial, but sometimes you need to round your answers for simplicity or because the problem specifically asks you to do so. Rounding to the nearest tenth is a common requirement when dealing with decimal numbers.

A few simple steps can help you carry out this rounding:

• Look at the digit in the hundredths place of your number.
• If this digit is 5 or greater, round up the tenths digit by 1.
• If it's less than 5, keep the tenths digit the same.

For instance, \(8.61\) becomes \(8.6\) when rounded to the nearest tenth. Rounding makes it easier to work with numbers, especially in practical applications where overly precise numbers are not necessary.