Problem 42 - Precalculus Mathematics for Calculus

Question

Solve the equation for the indicated variable. $$S=\frac{n(n+1)}{2} ; \text { for } n$$

Step-by-Step Solution

Answer

\( n = \frac{-1 \pm \sqrt{1 + 8S}}{2} \) is the solution for \( n \).

Step 1: Identify the equation and the target variable

The given equation is \( S = \frac{n(n+1)}{2} \), and we need to solve for the variable \( n \). This means we must express \( n \) in terms of \( S \) using algebraic manipulations.

Step 2: Eliminate the fraction

Multiply both sides of the equation by 2 to eliminate the fraction. This gives us \( 2S = n(n+1) \).

Step 3: Expand the equation

Expand the right side of the equation. Multiply \( n(n+1) \), which results in \( n^2 + n \). Thus, the equation becomes \( 2S = n^2 + n \).

Step 4: Rearrange into a quadratic equation

Rearrange the equation to bring all terms to one side, forming a standard quadratic equation. This gives us \( n^2 + n - 2S = 0 \).

Step 5: Solve the quadratic equation using the quadratic formula

The general form for solving a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = 1 \), and \( c = -2S \). Substitute these values into the quadratic formula: \( n = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-2S)}}{2 \cdot 1} \).

Step 6: Simplify the solution

Simplify within the square root: \( n = \frac{-1 \pm \sqrt{1 + 8S}}{2} \). This expression gives the possible values for \( n \).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula allows us to find the variable values even when factoring is not straightforward.

The quadratic formula is expressed as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

The formula requires knowing the coefficients \( a \), \( b \), and \( c \) from your equation, where:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

The expression under the square root, \( b^2 - 4ac \), is called the discriminant. It tells us about the nature of the solutions:

• If the discriminant is positive, there are two real and distinct solutions.
• If it is zero, there is exactly one real solution (or a repeated root).
• If it is negative, the solutions are not real (they are complex).

By plugging these coefficients into the quadratic formula, you can determine the possible solutions for your variable.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to isolate a specific variable. It's a crucial part of solving equations, especially when dealing with equations like the given one, \( S = \frac{n(n+1)}{2} \).

Here's how algebraic manipulation was applied:

• **Eliminating Fractions:** To simplify the work, the equation was multiplied by 2 to remove the fraction, resulting in \( 2S = n(n+1) \).
• **Expanding Expressions:** When dealing with a product term \( n(n+1) \), expansion helps to separate the terms: \( n^2 + n \).
• **Rearranging Terms:** The terms are brought together to form a quadratic equation like \( n^2 + n - 2S = 0 \). This format is pivotal for using the quadratic formula.

These steps help in setting up the problem for a straightforward application of the quadratic formula. Being comfortable with algebraic manipulation is key to solving complex equations efficiently.

Solving for Variables

Solving for a variable means expressing it explicitly in terms of the other quantities. This task is especially important when dealing with equations involving sums, products, or both, as seen in the problem \( S = \frac{n(n+1)}{2} \).

The process involves:

• **Isolating the Variable:** Through algebraic manipulations, modify the equation until the target variable stands alone on one side. It often involves combining like terms and ensuring all terms involving the desired variable are extracted or simplified properly.
• **Applying the Quadratic Formula:** Once isolated as part of a quadratic expression (like \( n^2 + n - 2S = 0 \)), using the quadratic formula will yield direct solutions for the variable \( n \).
• **Simplification:** After applying the quadratic formula, further simplification may be possible, such as reducing square roots or combining terms to find the most concise form of the solution. For example, \( n = \frac{-1 \pm \sqrt{1 + 8S}}{2} \).

By focusing on these steps, solving for variables becomes more systematic, ensuring that no critical stages are overlooked in achieving the correct solution.