Problem 33 - Algebra for College Students

Question

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$8 n^{2}-47 n-6=0$$

Step-by-Step Solution

Answer

The solutions are \( n = -\frac{1}{8} \) and \( n = 6 \).

Step 1: Identify the quadratic equation

The problem provides us with a quadratic equation, which is in the form \( an^2 + bn + c = 0\). Here, \( a = 8 \), \( b = -47 \), and \( c = -6 \).

Step 2: Apply the AC method

Calculate \( ac = 8 \times (-6) = -48 \). Look for two numbers that multiply to \(-48\) and add up to \(-47\). These numbers are \(-48\) and \(1\).

Step 3: Rewrite the middle term

Use the numbers found to decompose the middle term \(-47n\) into \(-48n + 1n\). The equation becomes \(8n^2 - 48n + n - 6 = 0\).

Step 4: Factor by grouping

Group the terms: \((8n^2 - 48n) + (n - 6)\). Factor the common factor out of each group: \(8n(n - 6) + 1(n - 6)\).

Step 5: Factor out the common binomial factor

The binomial \((n - 6)\) is common in both groups, so factor it out: \((8n + 1)(n - 6) = 0\).

Step 6: Solve for the roots

Set each factor equal to zero and solve for \( n \). For \( 8n + 1 = 0 \), we get \( n = -\frac{1}{8} \). For \( n - 6 = 0 \), we get \( n = 6 \).

Key Concepts

Factoring Techniques

Factoring is a key method for solving quadratic equations. The process involves rewriting a quadratic expression as a product of simpler expressions or factors.
By finding these factors, we make it easier to solve the equation. Factoring techniques focus on rearranging and simplifying equations to find their roots. This method is highly effective and is often necessary when dealing with complex quadratic equations.

There are various factoring methods, such as:

• Common Factor: Taking out a common factor that appears in each term.
• Difference of Squares: Using the formula \(a^2 - b^2 = (a + b)(a - b)\).
• Trinomial Factoring: Breaking down trinomials using different strategies, like the AC method.

Using the right technique can simplify the process and lead to an accurate solution.

AC Method

The AC method is a strategic approach specifically for factoring trinomials like the one in the quadratic form \(an^2 + bn + c = 0\). This method is useful when the coefficient \(a\) in the equation is not equal to 1, making it a preferred choice in many situations.

To use the AC method, you should:

• First, multiply \(a\) and \(c\), the coefficients of \(n^2\) and the constant term.
• Find two numbers that multiply to \(ac\) and add up to \(b\), the coefficient of \(n\).
• Use these numbers to break down the middle term \(bn\) into two terms that will be easier to factor.

This technique transforms complicated trinomials into expressions that can be tackled by other factoring methods, such as grouping, making the solution process simpler.

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials by grouping terms with common factors.
It is an effective way to simplify the process and solve the equation faster. Once you've used the AC method to split the middle term, factoring by grouping can be applied.

The steps to factor by grouping include:

• First, group the terms into pairs.
• Identify and factor out the greatest common factor (GCF) in each group.
• Ensure that the binomial factor appears in both groups to combine them into a single product.

This approach leverages the simplicity of factoring smaller groups rather than the entire polynomial at once, making it accessible and straightforward to solve quadratics.

Solving Quadratic Equations

Solving quadratic equations involves finding the values for the variable that satisfy the equation, meaning the values that make the equation true.
There are several methods to solve these equations, such as factoring, completing the square, and using the quadratic formula.

Using factoring methods, solutions can be found by:

• First, setting the factored expression equal to zero.
• Then, solving for the variable by solving each linear factor independently.

For example, in equations like \((8n + 1)(n - 6) = 0\), we set each factor to zero:\(8n + 1 = 0\) and \(n - 6 = 0\). Solving these gives the solutions \(n = -\frac{1}{8}\) and \(n = 6\).

Each solution process essentially breaks down complex quadratic equations into simpler linear equations, providing a clear pathway to the solutions.