Problem 39 - Prealgebra

Question

Solve each equation. Round to the nearest tenth, if necessary. $$2 d^{2}=162$$

Step-by-Step Solution

Answer

The solutions are \( d = 9 \) and \( d = -9 \).

Step 1: Isolate the variable term

Start by dividing both sides of the equation by 2 to isolate the term containing the variable: \( d^2 = \frac{162}{2} \). This simplifies to \( d^2 = 81 \).

Step 2: Apply the square root

To solve for \(d\), take the square root of both sides: \( d = \sqrt{81} \).

Step 3: Simplify the square root

Calculate the square root: \( \sqrt{81} = 9 \). Therefore, \( d = 9 \) or \( d = -9 \), since both positive and negative values satisfy the equation when squared.

Step 4: Confirm the solution range

Since the problem doesn't specify restrictions on \( d \), consider both positive and negative solutions as valid. However, given a typical context where only positive solutions are relevant (like distance), the positive solution would be \( d = 9 \) if specified.

Key Concepts

Isolate the Variable

In the process of solving equations, the first step often involves isolating the variable. This means getting the variable you are solving for by itself on one side of the equation. Here, the equation is initially given as \(2d^2 = 162\). To isolate the term \(d^2\), divide both sides by the coefficient of \(d^2\), which is 2.

• Divide both sides of the equation by 2: \(d^2 = \frac{162}{2}\)
• Simplify the division: \(d^2 = 81\)

Now, \(d^2\) is isolated, which simplifies the process of solving for \(d\). Isolating the variable is crucial because it prepares the equation for subsequent operations, like applying methods to find the value of the variable.

Square Root Method

The square root method is a common technique used when solving quadratic equations where the variable is squared. This method involves taking the square root of both sides of the equation once the variable term is isolated.

• For \(d^2 = 81\), take the square root of both sides to solve for \(d\): \(d = \sqrt{81}\).
• This simplifies to \(d = 9\), but we must remember both possible square roots: positive and negative.

The square root method transforms a quadratic equation into a linear one that is easier to solve. It is particularly useful for equations like \(x^2 = a\), where \(x\) is the variable we seek to determine.

Simplifying Equations

Simplifying equations is the process of making them as straightforward as possible. This involves performing operations that reduce the equation to its simplest form, facilitating easier solutions. In our example, by dividing \(2d^2 = 162\) by 2, we make the equation easier to handle: \(d^2 = 81\).

• Remove coefficients by dividing where applicable.
• Perform operations to get rid of complex terms or fractions.

Simplified equations mean less room for errors, as fewer steps are involved. It enhances clarity and guides you directly to the solution, reducing the chances of mistakes commonly made in more complex forms.

Positive and Negative Solutions

When solving quadratic equations, it is essential to remember that both positive and negative solutions can exist. This is because squaring either a positive or a negative number results in a positive number. For the equation \(d^2 = 81\), both \(d = 9\) and \(d = -9\) are valid solutions.

• \(d = 9\) because \(9^2 = 81\)
• \(d = -9\) because \((-9)^2 = 81\)

Understanding the existence of both forms of solutions is crucial, especially in applications like physics or engineering where both could be meaningful or could have different interpretations such as distance where negative values may not be applicable. Always check the context of the problem to determine whether both solutions are valid or if only the positive solution should be considered.