Problem 58 - Elementary and Intermediate Algebra

Question

Solve each proportion. $$ \frac{2}{x+6}=\frac{-2 x}{5} $$

Step-by-Step Solution

Answer

The solutions are \( x = -1 \) and \( x = -5 \).

Step 1: Identify the Problem

We are given the proportion \( \frac{2}{x+6} = \frac{-2x}{5} \). We need to find the value of \( x \) that makes this equation true.

Step 2: Cross Multiply

To solve the proportion, we cross multiply to eliminate the fractions. This gives us the equation: \[ 2 \cdot 5 = -2x \cdot (x + 6) \].

Step 3: Simplify the Equation

Perform the multiplications: \[ 10 = -2x(x + 6) \].

Step 4: Expand and Rearrange

Expand the right side and rearrange: \[ 10 = -2x^2 - 12x \].

Step 5: Set Equation to Zero

Move everything to one side of the equation to set it to zero: \[ 2x^2 + 12x + 10 = 0 \].

Step 6: Simplify the Coefficients

Divide the entire equation by 2 to simplify: \[ x^2 + 6x + 5 = 0 \].

Step 7: Factor the Quadratic

Factor the quadratic equation: \[ (x + 1)(x + 5) = 0 \].

Step 8: Solve for x

Setting each factor equal to zero gives us: \[ x + 1 = 0 \text{ or } x + 5 = 0 \]. Solving these gives the solutions: \[ x = -1 \text{ or } x = -5 \].

Key Concepts

Cross Multiplication

Cross multiplication is a fundamental technique useful for solving proportions. When you have an equation in the form of a proportion, such as \( \frac{a}{b} = \frac{c}{d} \), cross multiplication involves multiplying across the equals sign diagonally to eliminate the fractions. This method transforms the equation into \( a \cdot d = b \cdot c \).

For example, in the proportion \( \frac{2}{x+6} = \frac{-2x}{5} \), applying cross multiplication you get:

• Multiply 2 by 5, resulting in 10.
• Multiply \( -2x \) by \( (x+6) \), resulting in \( -2x(x+6) \).

This approach simplifies the expression, making it equation-based rather than fraction-based. Thus, it becomes easier to handle, like other algebraic equations. Cross multiplication is a reliable way to clear proportions and tackle the resulting equation swiftly.

Quadratic Equations

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. These equations are important in algebra as they model various natural phenomena and problem types.

In our exercise, after cross multiplying and simplifying, the equation \( 2x^2 + 12x + 10 = 0 \) was obtained. It is an example of a quadratic equation, where each term contributes to a specific characteristic:

• \( ax^2 \) is the quadratic term, influencing the parabola's width.
• \( bx \) is the linear term, affecting parabola orientation.
• \( c \) is the constant term, determining the y-intercept.

Quadratic equations can be solved by various methods, such as factoring, using the quadratic formula, or completing the square. In our exercise, solving it by factoring was more straightforward due to the equation's form.

Factoring Quadratics

Factoring quadratic equations is a method used to find the solutions of these equations by expressing them as a product of simpler factors. To factor a quadratic equation like \( x^2 + 6x + 5 = 0 \), we look for two numbers that multiply to the constant term (in this case, 5) and add to the linear coefficient (here, 6).

Through factoring, this equation becomes \( (x + 1)(x + 5) = 0 \).

This method leverages the zero product property, which states that if the product of two numbers is zero, then at least one of the numbers must be zero. Thus, solving \( (x + 1) = 0 \) gives \( x = -1 \), and solving \( (x + 5) = 0 \) gives \( x = -5 \).

Factoring is effective for quadratics that can be easily decomposed this way. However, if the quadratic is complex, the quadratic formula or completing the square might be better alternatives.