Problem 19 - Algebra for College Students

Question

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$-6 n^{2}+13 n-2=0$$

Step-by-Step Solution

Answer

The solutions are \(n = \frac{1}{6}\) and \(n = 2\).

Step 1: Identify an Appropriate Factoring Technique

The equation is a quadratic equation in the form \(an^2 + bn + c = 0\). We need to find two binomials whose product equals \(-6n^2 + 13n - 2\). Start by checking if it can be factored by looking for two numbers that multiply to \(a \times c = (-6) \times (-2) = 12\) and add up to \(b = 13\).

Step 2: Find Suitable Numbers for Factoring

We are looking for two numbers that multiply to 12 and add up to 13. These numbers are 12 and 1 because \(12 \times 1 = 12\) and \(12 + 1 = 13\).

Step 3: Re-write Middle Term Using the Suitable Numbers

We can rewrite the equation as \(-6n^2 + 12n + n - 2 = 0\). Here, we've split \(13n\) into \(12n + n\) to use our found numbers.

Step 4: Factor by Grouping

Group the terms as \([-6n^2 + 12n]\) and \([n - 2]\). Then, factor out the greatest common factor in each group: \(-6n(n - 2) + 1(n - 2)\). Notice that \((n - 2)\) is a common factor.

Step 5: Factor Out the Common Binomial

Factor \((n - 2)\) out of \(-6n(n - 2) + 1(n - 2)\) to get: \((-6n + 1)(n - 2) = 0\).

Step 6: Solve for the Variable

Now, set each factor equal to zero to solve for \(n\). First, set \(-6n + 1 = 0\), which gives \(-6n = -1\) and \(n = \frac{1}{6}\). Then, set \(n - 2 = 0\) which gives \(n = 2\).

Step 7: Write Final Solution

The solutions to the equation \(-6n^2 + 13n - 2 = 0\) are \(n = \frac{1}{6}\) and \(n = 2\).

Key Concepts

Factoring Techniques

Factoring techniques are essential tools for solving quadratic equations by breaking them down into simpler parts. Factoring involves finding two or more expressions that multiply to give the original quadratic equation. The process begins by identifying a quadratic equation in the standard form, which is \( ax^2 + bx + c = 0 \).

A key step is to look for two numbers that both multiply to the product of the leading coefficient \(a\) and the constant term \(c\), and also add up to the linear coefficient \(b\). This is known as the AC method.

• Step 1: Identify \(a\), \(b\), and \(c\) in the quadratic equation.
• Step 2: Multiply \(a\) and \(c\).
• Step 3: Find two numbers that multiply to that product and add to \(b\).

Once these numbers are found, you can rewrite the equation into a form that makes it easier to identify common factors.
Finally, group terms and extract common factors from each group, resulting in binomials that can be solved separately.

Quadratic Formula

The quadratic formula is a method used to find the solutions of a quadratic equation when factoring is not feasible. The formula provides a direct way to calculate the roots of any quadratic equation \( ax^2 + bx + c = 0 \).

The quadratic formula is given by:\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here is a step-by-step approach to using the quadratic formula:

• Step 1: Identify \(a\), \(b\), and \(c\) from the equation.
• Step 2: Calculate the discriminant \(b^2 - 4ac\). This term will determine the nature of the solutions.
• Step 3: Substitute \(a\), \(b\), and the discriminant into the formula.
• Step 4: Solve for \(n\) using the "\(\pm\)" sign, which indicates that there can be two solutions.

The quadratic formula is very powerful because it always provides the roots, whether they are real or complex.

Polynomial Equations

Polynomial equations are expressions that involve variables raised to whole number powers and coefficients. The degree of a polynomial equation is determined by the highest power of the variable it contains.

Quadratic equations, like \(-6n^2 + 13n - 2 = 0\), are a specific type of polynomial equation where the highest power is 2. Solving these involves techniques such as factoring and using the quadratic formula.

• Degree: The degree tells us the maximum number of solutions or roots the polynomial could have.
• Coefficients: These are the numbers multiplying the variable terms.
• Standard Form: Arranging the polynomial with terms in descending power order helps identify coefficients.

Understanding these components is critical because they influence the chosen method to solve the equation. Factoring is often preferred when possible for simplicity, but the quadratic formula can solve all quadratic polynomial equations reliably.