Problem 42 - Algebra for College Students

Question

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$n(n+1)=182$$

Step-by-Step Solution

Answer

The solutions are \( n = 13 \) and \( n = -14 \).

Step 1: Expand the Equation

The given equation is \( n(n+1) = 182 \). To simplify, expand the left side of the equation: \( n(n+1) = n^2 + n \). Thus, the equation becomes \( n^2 + n = 182 \).

Step 2: Rearrange the Equation

Subtract 182 from both sides to set the equation to zero: \( n^2 + n - 182 = 0 \). This represents a quadratic equation in the standard form \( ax^2 + bx + c = 0 \) where \( a = 1, b = 1, c = -182 \).

Step 3: Factor the Quadratic Equation

Look for two numbers that multiply to \(-182\) and add up to \(1\). The numbers \(13\) and \(-14\) satisfy this condition because \(13 imes (-14) = -182\) and \( 13 + (-14) = -1\). Thus, the factors of the quadratic are \((n + 14)(n - 13) = 0\).

Step 4: Solve for n

Set each factor equal to zero and solve for \( n \): \( n + 14 = 0 \) or \( n - 13 = 0 \). Solving these, we get \( n = -14 \) or \( n = 13 \).

Step 5: Verify the Solutions

Substitute the solutions back into the original equation to verify:For \( n = -14 \): \((-14)(-14+1) = 182 \rightarrow 182 = 182\). For \( n = 13 \): \(13(13+1) = 182 \rightarrow 182 = 182\).Both solutions satisfy the original equation.

Key Concepts

Factoring Techniques

Factoring is an essential mathematical tool, especially in solving quadratic equations. It involves expressing an algebraic expression as a product of simpler terms or factors. In the context of our quadratic equation, the focus is on identifying correct pairs of factors. These factors are needed to simplify or express the equation in its factored form, which makes it easier to solve.

To successfully factor a quadratic equation, you must determine two numbers that multiply to the product of the coefficient of the quadratic term and the constant (in this case, -182). These same numbers must also add up to the coefficient of the linear term (in this case, 1).

• Start by writing the quadratic equation in standard form: \(ax^2 + bx + c = 0\).
• Identify the values of \(a, b,\) and \(c\).
• Identify potential pairs of numbers that could be factors of the quadratic term.
• Use trial and error to find the correct pair that satisfies both the product and sum condition.

This method simplifies the equation into two binomial expressions, ensuring that each factor either increases or decreases the root of the equation depending on its form.

Quadratic Equations

Quadratic equations are a central topic in algebra. These are polynomial equations that involve terms up to the power of two and generally take the form \(ax^2 + bx + c = 0\). Understanding and solving quadratic equations is crucial as they are applicable in various fields such as physics, engineering, and finance.

The primary characteristic of a quadratic equation is the presence of a squared term, making it non-linear. This non-linearity is what makes solving them more complex than linear equations. Techniques used to solve quadratic equations must often transform or simplify the equation into a more manageable form.

• The standard form is the most beneficial for applying algebraic solutions such as factoring, completing the square, or using the quadratic formula.
• Symmetry is a key property; for example, the "parabolic" shape often associated with graphed quadratic equations.
• Solutions to quadratic equations, often called "roots," can be real or complex numbers.

To find these roots, we rely heavily on approaches that take advantage of the equation's structural properties.

Algebraic Solutions

Algebraic solutions refer to methods used to solve equations using algebraic manipulations. In solving the quadratic equation \(n^2 + n - 182 = 0\), algebraic solutions involve both expanding and rearranging the equation, allowing us to factor easily and solve for its roots.

To derive an algebraic solution, you must rearrange the equation to isolate variables and reduce complexities:

• Expand and simplify the equation to facilitate factoring.
• Use logical operations such as adding, subtracting, multiplying, or dividing both sides of the equation.
• Set each factor equal to zero to solve for the unknowns.

Once factored, each bracketed expression is equal to zero, leading us to two potential solutions for \(n\). This rule follows from the "Zero Product Property," which states that if a product is zero, at least one of the multiplicands must be zero. Verification of these solutions by substituting back into the original equation is necessary to ensure they satisfy the equation.