Problem 45 - Elementary and Intermediate Algebra

Question

Solve each proportion. $$ \frac{2}{3 x}=\frac{x}{6} $$

Step-by-Step Solution

Answer

The solutions are \( x = 2 \) or \( x = -2 \).

Step 1: Cross Multiply

To solve the proportion \( \frac{2}{3x} = \frac{x}{6} \), start by cross multiplying the terms. This means multiplying the numerator of one side by the denominator of the other side, and setting them equal. Thus, we have: \[ 2 \times 6 = 3x \times x \] This simplifies to: \[ 12 = 3x^2 \]

Step 2: Simplify the Equation

Now, divide both sides of the equation by 3 to solve for \( x^2 \): \[ \frac{12}{3} = \frac{3x^2}{3} \] This simplifies to: \[ 4 = x^2 \]

Step 3: Solve for x

To find \( x \), take the square root of both sides of the equation. Since \( x^2 = 4 \), we have: \[ x = \pm \sqrt{4} \] The solutions are: \[ x = 2 \quad \text{or} \quad x = -2 \]

Key Concepts

Cross Multiplication

Cross multiplication is a technique used to solve equations involving proportions, which are fractions that express a relationship of equality. When you are presented with a proportion like \( \frac{2}{3x} = \frac{x}{6} \), cross multiplication helps you eliminate the fractions to simplify the equation.
To perform cross multiplication, you multiply the numerator of one fraction by the denominator of the other fraction.

• For our example, this means multiplying \(2\) by \(6\), giving \(12\).
• Simultaneously, multiply \(3x\) by \(x\), yielding \(3x^2\).

Set these results equal to one another: \( 12 = 3x^2 \).
After cross multiplication, you'll typically have a simplified algebraic equation that's easier to solve. This method is particularly useful for maintaining balance in the equation, as it avoids the complication of fractions during initial algebraic manipulation.

Quadratic Equation

A quadratic equation is a polynomial equation in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( x \) represents an unknown. In our case, after cross-multiplying the proportion equation, we end up with: \( 3x^2 = 12 \).
This is a simplified form of a quadratic equation where \( a = 3 \), \( b = 0 \), and \( c = -12 \) when rewritten as \( 3x^2 - 12 = 0 \). Solving quadratic equations typically involves:

• Factoring the equation, although for simpler equations like \( 3x^2 = 12 \), direct manipulation is often used.
• Using methods such as the quadratic formula, but again, simpler forms allow for direct simplification.

In this scenario, divide both sides by 3 to isolate \( x^2 \), which gives: \( x^2 = 4 \).
Quadratic equations can yield two solutions because of the \( x^2 \) term. In our solution, finding \( x^2 = 4 \) leads to further steps involving square root calculation.

Square Root Calculation

Calculating the square root is a straightforward process, especially when you aim to solve equations like \( x^2 = 4 \). The principle here is to find a number that, when squared, equals the value under the square root sign.
This involves taking the square root of both sides of the equation. So,

• The square root of \( x^2 \) is \( x \).
• The square root of \( 4 \) is \( 2 \) because \( 2 \times 2 = 4 \).

Remember, square roots can produce both positive and negative results. Therefore, \( x = \pm2 \).
This reflects the dual nature of quadratic solutions, where two potential solutions exist: one positive and one negative.
It's important to compute correctly and recognize that roots provide all valid solutions to the original quadratic equation, making square numbers easy to work with in proportion problems like this one.