Problem 8 - Intermediate Algebra

Question

Solve each of the quadratic equations by factoring and applying the property, \(a b=0\) if and only if \(a=0\) or \(b=0\). If necessary, return to Chapter 3 and review the factoring techniques presented there. $$ 4 n^{2}+13 n=0 $$

Step-by-Step Solution

Answer

The solutions are \(n = 0\) and \(n = -\frac{13}{4}\).

Step 1: Identify a Common Factor

Look at the equation \(4n^2 + 13n = 0\). Notice that both terms share an \(n\) as a common factor. This indicates that we can factor \(n\) out of each term.

Step 2: Factor Out the Common Term

Factor \(n\) out of the expression. This changes the equation from \(4n^2 + 13n = 0\) to \(n(4n + 13) = 0\).

Step 3: Apply the Zero Product Property

According to the zero product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\). Here, \(a = n\) and \(b = 4n + 13\).

Step 4: Solve Each Factor for Zero

Set each factor equal to zero: \(n = 0\) and \(4n + 13 = 0\).

Step 5: Solve the Equation \(n = 0\)

The solution to \(n = 0\) is straightforward: \(n = 0\). No further work is required.

Step 6: Solve the Equation \(4n + 13 = 0\)

Subtract 13 from both sides to isolate the term with \(n\): \(4n = -13\).

Step 7: Divide to Solve for \(n\)

Divide both sides by 4 to solve for \(n\): \(n = -\frac{13}{4}\).

Step 8: State All Solutions

The equation \(4n^2 + 13n = 0\) has two solutions: \(n = 0\) and \(n = -\frac{13}{4}\).

Key Concepts

Factoring Techniques

When solving quadratic equations, factoring is a powerful technique that simplifies expressions and helps in finding solutions. Factoring is the process of breaking down an expression into simpler "factors" that can be multiplied together to get the original expression.
A common strategy for factoring involves identifying the greatest common factor (GCF) of all the terms in your equation. In the given equation, \(4n^2 + 13n = 0\), both terms have a common factor of \(n\). This means we can factor \(n\) out, simplifying the equation to \(n(4n + 13) = 0\).

• Look for numbers or variables common to all terms.
• Factor them out to simplify the equation.

This reduces complexity and transforms the equation into a product of simpler expressions. Mastery of factoring is essential because it often reduces the chances of errors in calculations later on.

Zero Product Property

The zero product property is crucial for solving factored equations. It states that if you have a product of two factors that equals zero, then at least one of the factors must be zero. Mathematically, it is stated as: If \(ab = 0\), then \(a = 0\) or \(b = 0\).
In our problem, after factoring out \(n\) from \(4n^2 + 13n = 0\), we have \(n(4n + 13) = 0\).
This means:

• \(n = 0\)
• or
• \(4n + 13 = 0\)

Using the zero product property allows you to set each factor to zero and solve separately. This technique greatly simplifies finding values for \(n\) that satisfy the original equation.

Solving Equations

Solving equations involves finding the values of the variables that make the equation true. After using the zero product property to split our equation into \(n = 0\) and \(4n + 13 = 0\), we solve each part:
1. **For** \(n = 0\):
This solution is direct and requires no additional steps; \(n = 0\) is already solved.
2. **For** \(4n + 13 = 0\):

• Subtract 13 from both sides: \(4n = -13\).
• Divide by 4 to get: \(n = -\frac{13}{4}\).

These steps give us the solutions to the equation, \(n = 0\) and \(n = -\frac{13}{4}\). Solving step-by-step and checking your work ensures that you accurately find all solutions derived from the factored equation.