Problem 32 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Cross Multiply

To solve the proportion \( \frac{2}{x+6} = \frac{-2x}{5} \), we start by cross multiplying. This gives us: \[ 2 \times 5 = -2x \times (x + 6) \] Simplifying the left side, we have:\[ 10 = -2x(x + 6) \]

Step 2: Expand and Simplify

Next, we expand the right side of the equation:\[ 10 = -2x^2 - 12x \]

Step 3: Rearrange the Equation

Rearrange the equation to get all terms on one side, aiming for a standard quadratic equation form:\[ 2x^2 + 12x + 10 = 0 \]

Step 4: Simplify the Quadratic

Divide every term by 2 to simplify:\[ x^2 + 6x + 5 = 0 \]

Step 5: Factor the Quadratic

Now, factor the quadratic equation. We are looking for two numbers that multiply to 5 and add to 6. These numbers are 5 and 1. Thus, we can factor the equation as:\[ (x + 5)(x + 1) = 0 \]

Step 6: Solve for x

Apply the zero-product property, which states that if a product of two factors equals zero, at least one of the factors must be zero.\[ x + 5 = 0 \quad \text{or} \quad x + 1 = 0 \]Solving these gives:\[ x = -5 \quad \text{or} \quad x = -1 \]

Key Concepts

Cross Multiplication: A Problem-Solving Trick

Cross multiplication is a powerful tool used to solve proportions, which are equations stating that two ratios are equal. In the original exercise, we had the proportion \( \frac{2}{x+6} = \frac{-2x}{5} \). To eliminate the fractions and make it easier to solve, we use cross multiplication. This technique involves multiplying the numerator of each fraction by the denominator of the opposite fraction.

In our example, cross multiply by first multiplying 2 by 5 to get 10, and then \(-2x\) by \(x+6\) to yield \(-2x(x+6)\). We’ve now transformed a fraction-based problem into a simpler algebraic equation: \(10 = -2x(x+6)\). Cross multiplication is especially useful because it helps us deal with linear or non-linear equations in a more straightforward manner by eliminating fractions outright. This method is a staple in algebra and a good foundational tool for solving equations.

Quadratic Equations: Unleashing Polynomial Power

A quadratic equation is any equation taking the form \( ax^2 + bx + c = 0 \) where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). They often arise in problems involving physics, business, and geometry. In step 3 of the original solution, we transformed our equation from cross multiplication into a standard quadratic format: \(2x^2 + 12x + 10 = 0\).

Quadratic equations can have up to two solutions, and these solutions may be real or complex numbers. Solving them involves arranging all the terms on one side (usually making one side zero) and simplifying, if possible. In our example, dividing through by 2 produced \(x^2 + 6x + 5 = 0\). Understanding quadratic equations allows students to tackle a vast array of mathematical problems efficiently.

Quadratics are not just academic exercises; they are also critically practical. For example, the trajectory of a basketball shot or the maximum area enclosed by a fence can all be modeled using quadratic equations. Therefore, mastering these equations opens up understanding in both theoretical and practical arenas.

Factoring Quadratics: Unlocking Solutions

Factoring quadratics is a technique used to solve quadratic equations by expressing them as a product of two binomial expressions. In the exercise, our simplified equation \(x^2 + 6x + 5 = 0\) led us to the factors \((x + 5)(x + 1) = 0\). To successfully factor, we needed to find two numbers that multiply to 5 (the constant term) and add up to 6 (the linear coefficient).

• The product: 5 \( \rightarrow \) is achieved by numbers 1 and 5
• The sum: 6 \( \rightarrow \) is obtained by adding 1 and 5

Thus, the factors of \(x^2 + 6x + 5\) were \(x + 5\) and \(x + 1\). Factoring is efficient because once a quadratic is written in its factored form, solving it becomes a matter of applying the zero-product property. This property dictates that if the product of two expressions is zero, at least one expression must be zero, leading directly to possible solutions \(x = -5\) and \(x = -1\). Factoring quadratics is a fundamental skill that not only helps in solving equations but also in understanding polynomial functions more deeply.