Problem 72 - College Algebra with Modeling and Visualization

Question

Solve the equation. Check your answers. $$ 2 n^{-2}-n^{-1}=3 $$

Step-by-Step Solution

Answer

The solutions are \( n = \frac{2}{3} \) and \( n = -1 \).

Step 1: Identify the Equation Format

The given equation is \( 2n^{-2} - n^{-1} = 3 \). Notice that it contains terms with negative exponents.

Step 2: Clear the Negative Exponents

Replace the negative exponents with their respective fractions. Rewrite the equation as \( \frac{2}{n^2} - \frac{1}{n} = 3 \).

Step 3: Find a Common Denominator

To combine the fractional terms, we need a common denominator. Here, it would be \( n^2 \). Rewrite the equation: \( \frac{2}{n^2} - \frac{n}{n^2} = 3 \).

Step 4: Combine the Fractions

Combine the fractional terms over the common denominator: \( \frac{2 - n}{n^2} = 3 \).

Step 5: Eliminate the Fraction

To remove the fraction, multiply both sides by \( n^2 \) to get \( 2 - n = 3n^2 \).

Step 6: Rearrange to a Quadratic Equation

Rearrange the equation to form a quadratic equation: \( 3n^2 + n - 2 = 0 \).

Step 7: Solve the Quadratic Equation

Use the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 3, b = 1, c = -2 \). Calculate: \( n = \frac{-1 \pm \sqrt{1 + 24}}{6} = \frac{-1 \pm 5}{6} \).

Step 8: Determine the Solutions

Calculate the two possible values for \( n \): \( n = \frac{4}{6} = \frac{2}{3} \) or \( n = \frac{-6}{6} = -1 \).

Step 9: Check the Solutions

Substitute \( n = \frac{2}{3} \) and \( n = -1 \) back into the original equation to verify. Both solutions satisfy the original equation: \( 2 \big( \frac{3}{2} \big)^2 - \big( \frac{3}{2} \big) = 3 \) and \( 2(-1)^{-2} - (-1)^{-1} = 3 \). Thus, both are correct.

Key Concepts

Understanding Negative Exponents

Negative exponents often confuse students at first, but the key is to understand that they are a way to express reciprocals. When you see a term like \( n^{-1} \), it translates to \( \frac{1}{n} \). Similarly, \( n^{-2} \) becomes \( \frac{1}{n^2} \). This transformation helps us work with equations more easily because it turns them into fractions, which can be manipulated using basic algebra.

Here are some helpful tips for working with negative exponents:

• A negative exponent means you divide 1 by that number raised to the positive of that exponent. For example, \( 5^{-3} = \frac{1}{5^3} = \frac{1}{125} \).
• Always convert all terms in an equation to avoid having negative exponents before proceeding with other steps. This makes the equation more manageable.
• When terms with negative exponents involve variables, try to express them as fractions over a common denominator to simplify the solving process.

By mastering the concept of negative exponents, you'll be well equipped to tackle more complex equations that incorporate these expressions.

Solving Quadratic Equations

Quadratic equations often appear intimidating, especially when they arise from rewriting exponents or fractions, as in the problem we're looking at. A standard quadratic equation is of the form \( ax^2 + bx + c = 0 \). In our exercise, we rearrange \( 2 - n = 3n^2 \) into \( 3n^2 + n - 2 = 0 \).

One of the most reliable methods of solving a quadratic equation is using the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here's a breakdown of the process:

• Identify coefficients: In this problem, \( a = 3 \), \( b = 1 \), and \( c = -2 \).
• Calculate the discriminant: Find \( b^2 - 4ac \). A positive result indicates real solutions. In our case, \( 1^2 - 4 \times 3 \times (-2) = 25 \).
• Apply the formula: Plug the values into the quadratic formula to find the solutions, which gives us \( n = \frac{-1 \pm 5}{6} \).

The solutions found are \( n = \frac{2}{3} \) and \( n = -1 \). These represent the values of \( n \) that make the original equation true.

Checking Solutions for Accuracy

Once you've found potential solutions to an equation, it's crucial to check them for accuracy. This involves substituting the solutions back into the original equation to see if they satisfy it.

For the equation we solved, \( 2n^{-2} - n^{-1} = 3 \), this means plugging \( n = \frac{2}{3} \) and \( n = -1 \) back into the equation:

• When \( n = \frac{2}{3} \): Rewrite the equation as \( 2(\frac{3}{2})^2 - (\frac{3}{2}) \). Simplify and you get the left side equal to 3.
• When \( n = -1 \): Replace the terms with negative exponents: \( 2(-1)^{-2} - (-1)^{-1} \). Simplify the expression and, again, you find it equals 3.

By successfully checking our solutions, we confirm their validity, ensuring that our calculations were correct. This step not only validates your work but also strengthens understanding and confidence in math skills.