Problem 19 - Beginning and Intermediate Algebra

Question

Solve. $$\frac{3 x}{x+2}+\frac{6}{x}=\frac{12}{x^{2}+2 x}$$

Step-by-Step Solution

Answer

The short answer based on the step-by-step solution is: The solutions for the equation \(\frac{3x}{x+2} + \frac{6}{x} = \frac{12}{x^2+2x}\) are \(x = 0\) and \(x = -2\).

Step 1: Find the common denominator

To combine fractions, we first need to find a common denominator. In this case, we notice that \(x^2+ 2x\) is the product of x and (x+2). The common denominator will be \(x(x+2)\).

Step 2: Make each fraction have the common denominator

Now, we will rewrite each of the three fractions in terms of the common denominator. We do this by multiplying both the numerator and denominator of each fraction by the terms necessary for the denominator to have a common denominator: \[\frac{3x}{x+2} \cdot \frac{x}{x} + \frac{6}{x} \cdot \frac{x+2}{x+2} = \frac{12}{x^2+2x}\]

Step 3: Simplify and combine the fractions

Now that we have found the common denominator, we can simplify the fractions and then combine them into a single fraction: \[\frac{3x^2}{x(x+2)} + \frac{6(x+2)}{x(x+2)} = \frac{12}{x(x+2)}\]

Step 4: Eliminate the denominator and simplify the equation

As the left side of the equation has the same denominator as the right side, we can eliminate the denominator and simplify the equation: \(3x^2 + 6(x+2) = 12\)

Step 5: Expand and solve the simplified equation

Now, we will expand and solve the simplified equation: \(3x^2 + 6x + 12 = 12\) \(3x^2 + 6x = 0\) To further simplify and solve the equation, we can factor it: \(3x(x + 2) = 0\) Now, we can find the values of x using the zero product property: \(3x = 0\) or \(x + 2 = 0\) The two possible solutions for x are: \(x = 0\) or \(x = -2\) Thus, the given equation has two solutions: \(x = 0\) and \(x = -2\).

Key Concepts

Fraction Operations

Fraction operations can often seem tricky due to the need for a common denominator when adding or subtracting fractions. In the exercise above, we begin by performing additions on fractions that do not have the same denominator. The key to solving these problems is to ensure each fraction shares a common denominator, which helps in combining them.
To find a common denominator:

• Identify the least common multiple of the denominators, which ensures all fractions align for easy addition or subtraction.
• For this equation, the denominators involve terms like \(x+2\) and \(x\), so we found that \(x(x+2)\) is the common denominator required.
• Ensure each fraction’s numerator and denominator are multiplied by necessary terms to achieve this common denominator.

By applying these steps, the fractions can be seamlessly combined, simplifying the equation.

Factoring Equations

Factoring is an essential algebraic skill that simplifies expressions and equations to make them more manageable. Often, equations arise from polynomial expressions that can be factored further.
In this exercise, after simplifying, we ended up with the equation \(3x^2 + 6x = 0\). Here, factoring helps to break down the expression:

• First, look for common factors in all terms. In \(3x^2 + 6x\), both terms share \(3x\) as a common factor.
• By factoring out \(3x\), the equation becomes \(3x(x + 2) = 0\).

Factoring not only makes the equation easier to solve but is particularly useful in employing the zero product property, which is the next step in solving the equation.

Zero Product Property

The Zero Product Property is a crucial concept when dealing with equations that have been factored into simpler forms. It states that if the product of two factors equals zero, then at least one of the factors must be zero.
After factoring \(3x(x + 2) = 0\):

• Apply the zero product property: Set each factor equal to zero. This gives the equations \(3x = 0\) and \(x + 2 = 0\).
• Solving \(3x = 0\) results in \(x = 0\).
• Solving \(x + 2 = 0\) results in \(x = -2\).

This method enables us to find solutions to equations precisely and quickly, confirming two potential solutions for the values of \(x\).

Rational Equations

Rational equations are equations that involve fractions whose numerators and/or denominators contain algebraic expressions. Solving such equations requires careful manipulation to clear fractions and isolate the variable.
In this exercise, we first equalled fractions by finding a common denominator and then simplified. Once fractions are cleared, it transforms into a simpler algebraic equation.

• Through steps of simplification, one must handle the equations as seen here: equivalence allows removal of shared denominators from both sides.
• From the step \(3x^2 + 6(x+2) = 12\), clearing the fraction eases solving, enabling clean factorization and application of the zero product property.

These techniques ensure precise solutions for unknown variables in rational equations, providing a structured approach that works through fraction complexities in algebra.