Problem 41 - Algebra for College Students

Question

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$n(n+2)=360$$

Step-by-Step Solution

Answer

The solutions are \(n = -20\) and \(n = 18\).

Step 1: Expand the expression

Start by expanding the left-hand side of the equation: \(n(n+2) = n^2 + 2n\). This equation now looks like \(n^2 + 2n = 360\).

Step 2: Move all terms to one side

Subtract 360 from both sides to set the equation to zero: \(n^2 + 2n - 360 = 0\). This is a quadratic equation that we will need to factor.

Step 3: Factor the quadratic expression

Look for two numbers that multiply to -360 and add to 2. These numbers are 20 and -18. Rewrite the equation as \((n + 20)(n - 18) = 0\).

Step 4: Solve for n

Set each factor equal to zero: \(n + 20 = 0\) or \(n - 18 = 0\). Solving these gives \(n = -20\) or \(n = 18\).

Step 5: Verify the solutions

Check both solutions in the original equation to ensure they are correct. Substitute back into \(n(n+2)=360\): For \(n = -20\), \(-20(-18) = 360\) and for \(n = 18\), \(18(20) = 360\), so both solutions are verified.

Key Concepts

Factoring Techniques

Factoring is a crucial technique in solving quadratic equations, like the one in our exercise. We often need to rewrite a trinomial into a product of two binomials. This makes it easier to solve the equation by finding the values of the variable that make each binomial zero.
To factor a quadratic equation, we look for two numbers that multiply to the constant term at the end of the polynomial and add up to the coefficient of the middle term.

• For example, in the equation from the exercise, which is in the form of \(n^2 + 2n - 360\), we need numbers that multiply to \(-360\) and add up to \(2\).
• After examination, we find \(20\) and \(-18\) fit these conditions.
• Therefore, we can rewrite the quadratic as \((n + 20)(n - 18) = 0\).

This approach simplifies solving, as we can then apply the zero product property to find the roots of the equation.

Solving Equations

Once we have factored the quadratic equation, solving it involves setting each factor equal to zero. This is known as the zero-product property, which states that if a product of factors equals zero, at least one of the factors must be zero.

• For our exercise, after factoring the equation to \((n + 20)(n - 18) = 0\), we take each factor separately:
• Set \(n + 20 = 0\) which leads to \(n = -20\).
• Set \(n - 18 = 0\) which results in \(n = 18\).

This step is crucial because it gives us potential solutions for the original equation. However, it’s important to always verify these solutions by plugging them back into the original equation.

Algebraic Expansion

Algebraic expansion is the process of multiplying out expressions and is an essential step in solving quadratic equations. By expanding, we transform a factorable binomial form into a standard quadratic form.
In the given exercise, we began with \(n(n+2) = 360\).

• By multiplying \(n\) and \((n+2)\), we expand it to \(n^2 + 2n\).
• This step allows us to see the quadratic form clearly: \(n^2 + 2n = 360\).

Understanding expansion is vital as it sets the stage for further manipulation like moving terms and eventually factoring to simplify and solve the equation.