Problem 13 - Algebra for College Students

Question

Solve each quadratic equation by using (a) the factoring method and (b) the method of completing the square. $$n(n+6)=160$$

Step-by-Step Solution

Answer

The solutions are \( n = 10 \) and \( n = -16 \).

Step 1: Write the Quadratic Equation in Standard Form

We start with the given equation \( n(n + 6) = 160 \). Expand the left side to get \( n^2 + 6n = 160 \). Then, subtract 160 from both sides to write the equation in standard form: \( n^2 + 6n - 160 = 0 \).

Step 2: Solve by Factoring

First, factor the quadratic equation \( n^2 + 6n - 160 = 0 \). We need two numbers whose product is -160 and sum is 6. These numbers are 16 and -10. Rewrite the equation as \( (n + 16)(n - 10) = 0 \).

Step 3: Apply the Zero Product Property

According to the zero product property, if \( (n + 16)(n - 10) = 0 \), then either \( n + 16 = 0 \) or \( n - 10 = 0 \). Solve each equation: \( n + 16 = 0 \Rightarrow n = -16 \) and \( n - 10 = 0 \Rightarrow n = 10 \).

Step 4: Solve by Completing the Square

Start with \( n^2 + 6n = 160 \). Move the constant term to the right side to focus on the quadratic and linear terms: \( n^2 + 6n = 160 \). Take half of the coefficient of \( n \), square it: \( \left( \frac{6}{2} \right)^2 = 9 \), and add to both sides: \( n^2 + 6n + 9 = 169 \).

Step 5: Express as a Binomial Square

The left side \( n^2 + 6n + 9 \) becomes \( (n + 3)^2 \). So, the equation is \( (n + 3)^2 = 169 \).

Step 6: Solve the Binomial Square Equation

Take the square root of both sides: \( n + 3 = \pm 13 \). Solve these: \( n + 3 = 13 \Rightarrow n = 10 \) and \( n + 3 = -13 \Rightarrow n = -16 \).

Step 7: Verify the Solutions

The solutions from both methods are \( n = 10 \) and \( n = -16 \). Substitute each back into the original equation \( n(n + 6) = 160 \) to verify correctness. For \( n = 10 \), we have \( 10(10 + 6) = 160 \). For \( n = -16 \), we have \( -16(-16 + 6) = 160 \). Both check out.

Key Concepts

Factoring Method

To solve a quadratic equation by factoring, you focus on breaking the quadratic expression into a product of two binomials. In the exercise provided, we started with the equation in standard form: \[ n^2 + 6n - 160 = 0 \]. The aim is to find two numbers that multiply to give the constant term, -160, and add up to the linear coefficient, which is 6.

• First find such numbers: 16 and -10 fit perfectly because 16 \( \times \) -10 equals -160, and 16 + (-10) equals 6.
• Thus, the quadratic can be factored as \( (n + 16)(n - 10) = 0 \).

After factoring, you apply the zero product property to find the roots of the equation. So we solve \( n + 16 = 0 \) and \( n - 10 = 0 \), leading to solutions \( n = -16 \) and \( n = 10 \). The factoring method is often the quickest way to solve simple quadratic equations, provided the quadratic expression is factorable.

Completing the Square

Completing the square is a method that helps to solve quadratic equations by turning the quadratic expression into a perfect square trinomial. Starting with the equation \( n^2 + 6n = 160 \), the first step is to bring the constant term over to the right side. To make the left side a perfect square, take half of the linear coefficient (6 in this case), square it, and then add to both sides.

• Half of 6 is \( \frac{6}{2} = 3 \) and \( 3^2 = 9 \).
• Add 9 to each side: \( n^2 + 6n + 9 = 160 + 9 \).
• You now have \( (n + 3)^2 = 169 \).

To solve, take the square root of both sides resulting in \( n + 3 = \pm 13 \). This gives \( n = 10 \) and \( n = -16 \). Completing the square is a valuable method, especially for solving quadratic equations that do not factor neatly or when deriving the quadratic formula.

Zero Product Property

The Zero Product Property is a fundamental concept used in solving quadratic equations. It states that if the product of two factors is zero, then at least one of the factors must be zero. In the context of the exercise, after factoring the quadratic equation to \( (n + 16)(n - 10) = 0 \), the property allows us to deduce:

• If \( n + 16 = 0 \), then \( n = -16 \).
• If \( n - 10 = 0 \), then \( n = 10 \).

This step is pivotal because it transforms the problem into solving simple linear equations. The zero product property is what makes factoring such an efficient method for solving quadratics. It directly leads us to the equation's solutions, emphasizing the importance of understanding and utilizing this property in algebra.