Problem 24 - Precalculus : Building Concepts and Connections

Question

Solve the rational equation. Check your solutions. $$\frac{1}{x^{2}}-\frac{7}{x}=18$$

Step-by-Step Solution

Answer

The solutions to the given rational equation are \(x = -0.0526, 1.009\).

Step 1: Find the common denominator

In order to solve the equation, it is easier to handle if all terms have the same denominator. The common denominator for these fractions is \(x^{2}\). Now, maintain this common denominator throughout the equation.

Step 2: Rewrite the equation

The equation now becomes \(1 - 7x = 18x^{2}\). This equation is resulted from multiplying the term \(-\frac{7}{x}\) by \(x\) to match the \(x^{2}\) in the denominator of the first term.

Step 3: Transform into a quadratic equation

Rearranging the terms will transform the equation into a standard quadratic equation. The new equation will be \(18x^{2} + 7x - 1 = 0\).

Step 4: Solve the quadratic equation

This equation can be solved by factoring, completing the square, or using the quadratic formula. In this case, we use quadratic formula. The solutions to the equation are then \(x = \frac{-7 \pm \sqrt{(7)^2 - 4.18.(-1)}}{2.18}\).

Step 5: Calculate the solutions

After calculating above expression, we get two roots as \(x = -0.0526, 1.009\).

Key Concepts

Quadratic Equations

Before dealing with eigenvectors and eigenvalues, another interesting mathematical phenomenon occurs: quadratic equations. These are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The standard form makes it straightforward to handle these equations. For our specific exercise,
"transform the rational equation" into a quadratic form was key in solving it. With the equation \(18x^{2} + 7x - 1 = 0\), we put our given equation into a recognizable quadratic form. This
quadratic form facilitates using the quadratic formula to find solutions.

Common Denominator

When you're working with rational equations, finding a common denominator is crucial. It allows you to combine fractions into a single equation. Here, the original fractions \(\frac{1}{x^{2}}\) and \(\frac{7}{x}\) had different denominators: \(x^2\) and \(x\), respectively. By choosing a common denominator of \(x^2\), we align our terms to simplify the equation. This step is fundamental:

• Multiplies each term by \(x^2\) to clear the fractions.
• Results in a single equation without fractions to solve.
• Makes conversion to a quadratic form easier.

Without this method, subsequent steps like solving the quadratic equation would be far more complex.
Understanding and practicing finding a common denominator helps in efficiently solving rational equations.

Quadratic Formula

The quadratic formula is a powerful tool used when solving quadratic equations that cannot be easily factored. The formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In our exercise, after rewriting the rational equation as a quadratic equation, we utilized the quadratic formula to find possible values for \(x\). The parameters \(a = 18\), \(b = 7\), and \(c = -1\) are substituted into the formula, leading to the solutions \(x = -0.0526\) and \(1.009\).
The quadratic formula provides:

• Exact solutions for quadratic equations.
• Ways to understand the nature of the roots.
• Insights into whether solutions are real or complex.

This formula is an essential tool for anyone working with quadratic equations.

Checking Solutions

Checking your solutions to an equation is a crucial step. It confirms that the values you found actually satisfy the original equation. Often, when dealing with rational equations, some solutions may not be valid due to domain restrictions, like division by zero.Upon finding the solutions \(x = -0.0526\) and \(1.009\), you substitute them back into the original equation, \(\frac{1}{x^2} - \frac{7}{x} = 18\), to verify their validity. If substituting yields a true statement, then these solutions are correct. If not, a mistake may have occurred due to overlooked constraints or computational errors.Checking solutions is especially important:

• Ensures that both or neither of the roots are valid.
• Avoids errors from overlooking non-permissible values due to division.
• Increases confidence in your result.

This step is as important as finding solutions themselves, ensuring they are part of the domain of the equation.