Problem 21 - Algebra for College Students

Question

Solve each equation. $$n+\frac{3}{n}=\frac{19}{4}$$

Step-by-Step Solution

Answer

The solutions are \(n = 4\) and \(n = \frac{3}{4}\).

Step 1: Clear the Denominator

To eliminate the fraction, multiply every term by the common denominator, which is 4n. This gives us the equation: \(4n^2 + 12 = 19n\).

Step 2: Rearrange into a Standard Quadratic Equation

Rearrange all the terms to one side of the equation to form a standard quadratic equation: \(4n^2 - 19n + 12 = 0\).

Step 3: Use the Quadratic Formula

The quadratic formula is given by \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where a, b, and c are coefficients from \(ax^2 + bx + c = 0\). Here, \(a = 4\), \(b = -19\), and \(c = 12\). Substitute these values into the formula: \[n = \frac{19 \pm \sqrt{(-19)^2 - 4 \cdot 4 \cdot 12}}{2 \cdot 4}\].

Step 4: Calculate the Discriminant and Solve

Calculate the discriminant:\((-19)^2 - 4 \times 4 \times 12 = 361 - 192 = 169\).Use the discriminant to find the solutions:\[n = \frac{19 \pm \sqrt{169}}{8}\].

Step 5: Simplify the Roots

Since \(\sqrt{169} = 13\), substitute back:\[n = \frac{19 + 13}{8} = 4\]and \[n = \frac{19 - 13}{8} = \frac{3}{4}\].

Key Concepts

Quadratic Formula

When you encounter a quadratic equation, the quadratic formula is a versatile tool to find its solutions. A quadratic equation typically looks like \(ax^2 + bx + c = 0\). The quadratic formula helps find the values of \(x\) that satisfy this equation. It is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula can solve any quadratic equation provided you know the coefficients \(a\), \(b\), and \(c\). Here's how it works:

• Identify the coefficients \(a\), \(b\), and \(c\) in your equation.
• Substitute these values into the formula.
• Calculate the discriminant to determine the nature of the roots.

Discriminant

The discriminant is part of the quadratic formula and is key to understanding the nature of the roots of a quadratic equation. It is represented by \(b^2 - 4ac\). The value of the discriminant reveals the characteristics of the solutions:

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, the equation has exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, the equation has two complex roots.

For example, in our original equation, calculating the discriminant as \((-19)^2 - 4 \times 4 \times 12\), we find it is 169, a positive number. This informs us that there are two distinct real solutions.

Solving Equations

Solving a quadratic equation requires moving from the given form to finding the solutions, or roots. Begin by identifying the quadratic equation format \(ax^2 + bx + c = 0\) from your equation. Rearrange your equation if necessary, to bring all terms to one side of the equation.
Using the quadratic formula, input your coefficients \(a\), \(b\), and \(c\) into it. Calculating the roots involves solving the formula step-by-step:

• Calculate the discriminant.
• Use the discriminant in the quadratic formula.
• Compute the roots based on the formula’s results.

This approach ensures that you correctly address all needed calculations, leading to finding accurate solutions.

Simplifying Roots

Once you arrive at the expression for the roots using the quadratic formula, you might need to simplify the results. Simplification might include basic arithmetic or algebraic steps. For example, after solving using the formula, you may find results expressed with square roots or fractions. Recognizing a perfect square under a square root, like \(\sqrt{169}\), directly simplifies to 13.

• Calculate the exact numerical values from the roots.
• Reduce fractions whenever possible.
• Ensure all arithmetic steps are correct, giving precise final results like \(n = 4\) or \(n = \frac{3}{4}\).

This step ensures you present clean, understandable solutions that match mathematical standards.