Problem 43 - Elementary Algebra

Question

For the following problems, solve the rational equations. $$ \frac{2}{x^{2}}+\frac{1}{x}=1 $$

Step-by-Step Solution

Answer

Answer: The solutions to the given equation are \(x = 0\) and \(x = -1\).

Step 1: Find a common denominator

To find a common denominator between \(\frac{2}{x^2}\) and \(\frac{1}{x}\), we take their least common multiple (LCM) which would be \(x^2\) in this case.

Step 2: Add the fractions with the common denominator

To add the two fractions, we rewrite both fractions with \(x^2\) as the denominator: $$ \frac{2}{x^{2}}+\frac{x}{x^{2}}=\frac{2+x}{x^{2}} $$ Now we could rewrite the original equation as: $$ \frac{2+x}{x^{2}}=1 $$

Step 3: Eliminate the denominator

To eliminate the denominator, we will multiply both sides of the equation by \(x^2\). This gives us: $$ (2+x)(x^{2}) = (x^{2}) $$

Step 4: Simplify and solve for x

Distribute \(x^2\) on the left side of the equation: $$ 2x^{2} + x^{3} = x^{2} $$ Now subtract \(x^2\) from both sides to get: $$ x^{3} + x^{2} = 0 $$ Now, we can factor out an x: $$ x^2(x + 1) = 0 $$ Now we use the Zero Product Property, which states that if \(ab=0\), then either \(a=0\) or \(b=0\). In this case, $$ x^2 = 0 \Rightarrow x = 0 $$ or $$ x + 1 = 0 \Rightarrow x = -1 $$ Thus, the solutions to the given equation are \(x = 0\) and \(x = -1\).

Key Concepts

Common Denominator

When dealing with rational equations, a common denominator is crucial for simplifying terms and equations. It is the same denominator shared by two or more fractions, allowing you to combine them into a single term.
For example, in the equation \( \frac{2}{x^2} + \frac{1}{x} = 1 \), the denominators are different. To simplify, we need the least common multiple (LCM) of the denominators, which in this case is \( x^2 \).
Here is how finding a common denominator helps:

• Makes it easier to add or subtract fractions.
• Simplifies the equation to make further calculations more straightforward.

In our example, rewriting \( \frac{1}{x} \) with a denominator of \( x^2 \) involves multiplying the numerator and the denominator by \( x \), resulting in \( \frac{x}{x^2} \). The new equation becomes \( \frac{2+x}{x^2} = 1 \). Now, adding the fractions is a lot simpler!

Factoring

Factoring is an essential skill for solving polynomial equations and simplifying complex expressions. It involves breaking down a composite expression into simpler, more manageable parts.
In the solution provided, after eliminating the denominator, we simplified the equation to \( x^3 + x^2 = 0 \). This is where factoring comes into play.
Here's a simple guide on factoring:

• Identify any common factors in the terms.
• Factor out the greatest common factor (GCF). In the equation \( x^3 + x^2 = 0 \), the GCF is \( x^2 \).
• Rewrite the expression: \( x^2(x + 1) = 0 \)

Factoring makes solving equations straightforward by reducing them to simpler parts. Without factoring, you would have to solve a more complex polynomial, which can be overwhelming.

Zero Product Property

The zero product property is a core concept in algebra. It states that if the product of two expressions is zero, at least one of the expressions must be zero. This property is extremely useful for solving equations involving products set to zero.
Once we factor the expression \( x^2(x + 1) = 0 \), we can apply the zero product property:

• Look at each factor separately: \( x^2 = 0 \) or \( x + 1 = 0 \).
• If \( x^2 = 0 \), then \( x = 0 \).
• If \( x + 1 = 0 \), solve for \( x \) to find \( x = -1 \).

Once you have factored the expression and applied the zero product property, solving the equation becomes a matter of simple algebra. This allows you to find the solutions to the rational equation effectively.