Problem 37 - Algebra for College Students

Question

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$35 n^{2}-18 n-8=0$$

Step-by-Step Solution

Answer

The solutions are \(n = -\frac{2}{7}\) and \(n = \frac{4}{5}\).

Step 1: Identify the Type of Equation

We are given a quadratic equation in the form \(35n^2 - 18n - 8 = 0\). This is a quadratic equation and can be solved by factoring.

Step 2: Multiply and Find Two Numbers

To factor, we first need to multiply the coefficient of \(n^2\) (which is 35) by the constant term (which is -8). This gives us \(35 \times (-8) = -280\). Next, we need to look for two numbers whose product is -280 and whose sum is the coefficient of the linear term, which is -18.

Step 3: Choose the Correct Pair of Numbers

After checking the factor pairs, we find that the numbers -28 and 10 multiply to -280 and add to -18. So, we will use these two numbers to split the middle term of the quadratic equation.

Step 4: Rewrite the Middle Term

Rewrite the middle term \(-18n\) using -28 and 10 as \(-28n + 10n\). This transforms the equation to:\[35n^2 - 28n + 10n - 8 = 0\]

Step 5: Factor by Grouping

Group the terms into two pairs:\[(35n^2 - 28n) + (10n - 8) = 0\]Factor out the greatest common factor from each group:\[7n(5n - 4) + 2(5n - 4) = 0\]

Step 6: Factor Out the Common Binomial Factor

Notice that \((5n - 4)\) is a common factor. Factor it out:\[(7n + 2)(5n - 4) = 0\]

Step 7: Solve for n

Set each factor equal to zero and solve for \(n\):1. \(7n + 2 = 0\) - Subtract 2 from both sides: \(7n = -2\) - Divide by 7: \(n = -\frac{2}{7}\)2. \(5n - 4 = 0\) - Add 4 to both sides: \(5n = 4\) - Divide by 5: \(n = \frac{4}{5}\)

Key Concepts

Factoring Techniques

Factoring techniques play a crucial role in solving quadratic equations efficiently. One of the simplest and most effective methods of solving these types of equations is factoring, which involves expressing a polynomial as a product of simpler polynomials.

The basic idea of factoring is to rewrite the quadratic equation in a form that makes it easier to identify when the product is zero. For example, when you have a quadratic equation like \( ax^2 + bx + c = 0 \), factoring helps us break it down into two binomials that can be set to zero.

• To start, ensure that your quadratic is in standard form \( ax^2 + bx + c \).
• Look for two numbers that multiply to give the product of \( a \times c \) and add to give \( b \).
• Rewrite the middle term using these two numbers and factor by grouping.

Factoring requires practice, and while it can be a bit tricky to find the right pairs at first, over time, identifying these pairs becomes easier. Understanding these techniques can make solving quadratic equations much less daunting.

Solving Quadratic Equations

Solving quadratic equations is a fundamental concept in algebra. It involves finding the values of the variable that satisfy the equation \( ax^2 + bx + c = 0 \). Two popular methods for finding the solutions are factoring and using the quadratic formula.

When solving by factoring, as in our given equation of \( 35n^2 - 18n - 8 = 0 \), the ultimate goal is to express the quadratic as a product of two linear factors. Once factored, the principle of zero product property helps, which states if the product of two factors is zero, then at least one of the factors must be zero.

• Factor the quadratic equation.
• After factoring, you get expressions such as \((7n + 2)(5n - 4) = 0\).
• Set each factor equal to zero and solve for the variable.

This solution process provides two possible answers because a quadratic equation generally has two solutions. Through this method, one can determine the values of \( n \) in a systematic and straightforward manner.

Factoring by Grouping

Factoring by grouping is a strategic approach used to factor quadratic equations, especially when a straightforward factorization isn't apparent. It involves rearranging and grouping terms into pairs. Each group is then factored separately to reveal a common factor throughout the equation.

Here's how factoring by grouping works, as shown in the solution for \( 35n^2 - 18n - 8 = 0 \):

• First, identify two numbers that multiply to the product of the first coefficient and the constant and add up to the middle coefficient.
• Rewrite the equation, splitting the middle term into two terms using these numbers.
• Group the terms into two parts: \((35n^2 - 28n) + (10n - 8)\).
• Factor out the greatest common factor from each group.

In this example, the terms were grouped and simplified, leading to the common binomial \((5n - 4)\). This common binomial can be factored out, completing the factorization of the quadratic equation.

Factoring by grouping is a powerful technique that enriches your arsenal of problem-solving strategies for quadratic equations.