Problem 55 - Prealgebra

Question

Solve each equation. Round to the nearest tenth, if necessary. $$\frac{x^{2}}{2}=51$$

Step-by-Step Solution

Answer

\( x \approx \pm 10.1 \)

Step 1: Understand the Equation

We start with the equation \( \frac{x^{2}}{2}=51 \). This equation states that the square of \( x \) divided by 2 equals 51.

Step 2: Clear the Fraction

To eliminate the fraction, multiply both sides of the equation by 2. This gives us \( x^{2} = 51 \times 2 \).

Step 3: Simplify the Right Side

Calculate \( 51 \times 2 \) to simplify the right side of the equation, yielding \( x^{2} = 102 \).

Step 4: Solve for x by Taking the Square Root

To solve for \( x \), take the square root of both sides of the equation: \( x = \pm\sqrt{102} \).

Step 5: Calculate and Round the Square Root

Approximating \( \sqrt{102} \) gives \( 10.0995\ldots \). Since we need to round to the nearest tenth, the solution is approximately \( x \approx \pm 10.1 \).

Key Concepts

Understanding Squaring

Squaring is a mathematical operation where you multiply a number by itself. This action is denoted by the small "2" raised above the number, like this:

• For example, if you have a number \( x \), then squaring it means calculating \( x^2 = x \times x \).
• For instance, the square of 5 is \( 5^2 = 25 \).

In the context of solving equations, squaring plays a crucial role, especially in quadratic equations. When you encounter an equation like \( \frac{x^2}{2}=51 \), you're dealing with \( x \) squared. This means you're looking for a number \( x \) that, when squared and divided by 2, equals 51.Working through this kind of equation involves:

• Identifying the square in the equation.
• Using operations to isolate \( x^2 \) so you can later solve for \( x \).

Exploring Square Roots

A square root operation accomplishes the opposite of squaring. It finds a number which, when multiplied by itself, equals the original value under the square root:

• The square root of \( 25 \) is \( 5 \) because \( 5 \times 5 = 25 \).
• Similarly, \( \sqrt{49} = 7 \) because \( 7 \times 7 = 49 \).

In algebra, solving for \( x \) often involves these steps:

• First, you need to isolate the term \( x^2 \).
• Then take the square root of both sides to solve for \( x \).

When solving \( x^2 = 102 \), taking the square root of both sides yields \( x = \pm \sqrt{102} \). Remember:

• Square roots have both positive and negative values.
• For example, \( \sqrt{102} \approx 10.0995 \), so \( x \approx \pm 10.1 \) when rounded.

The Rounding Decimals Technique

Rounding decimals helps in simplifying a number to make it easier to read and use in real-world applications. We often round to a certain place value, such as the nearest tenth. Here's how you round decimal numbers:

• Look at the number in the place value just after the one you're rounding to.
• For example, with \( 10.0995 \), look at the number in the hundredths place: 9.
• If this number is 5 or greater, you round the tenths place up. If it's less than 5, leave it as is.

Using these guidelines:

• In solving \( \pm \sqrt{102} \approx 10.0995 \), you round to 10.1.
• The decision to round up is based on the second digit after the decimal in 10.0995.

Thus, knowing how to round decimals allows you to provide clear and precise answers, which is essential in mathematics and everyday calculations.