Problem 20 - Elementary Algebra

Question

Solve the following quadratic equations. \((n+5)^{2}=32\)

Step-by-Step Solution

Answer

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

Step 1: Expand the equation

To solve the quadratic equation \( (n+5)^{2}=32 \), start by taking the square root of both sides. This gives us: \( n+5 = \pm \sqrt{32} \).

Step 2: Simplify the square root

Simplify \( \sqrt{32} \) to \( \sqrt{16 \cdot 2} = 4 \sqrt{2} \). Therefore, the equation becomes: \( n+5 = \pm 4 \sqrt{2} \).

Step 3: Solve for n

Solve for n by isolating it on one side: \( n = \pm 4 \sqrt{2} - 5 \). This results in two possible solutions: \( n = 4 \sqrt{2} - 5 \) and \( n = -4 \sqrt{2} - 5 \).

Key Concepts

Taking Square Roots

When you encounter a quadratic equation in the format \((x + a)^2 = b\), the first step is to take the square root of both sides. This action helps to simplify the equation and makes it easier to solve. By taking the square root of both sides of our example equation \((n + 5)^2 = 32\), we directly simplify to the next step: \(n + 5 = \pm \sqrt{32}\).
Notice the ± symbol; it indicates that there are two possible values after taking the square root: the positive and the negative square root. This is crucial because quadratic equations typically have two solutions. Remember, always apply the square root to both sides of the equation and include the ± symbol to capture all solutions.

Simplifying Radicals

Once we have our equation in a simpler form \(n + 5 = \pm \sqrt{32}\), the next step is to simplify the square root. Simplifying radicals involves breaking down the number inside the square root into its prime factors and finding pairs of the same number.
For \(\sqrt{32}\), we can rewrite 32 as \(16 \cdot 2\). Since 16 is a perfect square (\(4^2\)), we can simplify further: \(\sqrt{32} = \sqrt{16 \cdot 2} = 4 \sqrt{2}\). This simplification makes our equation easier to handle: \(n + 5 = \pm 4 \sqrt{2}\).
Remember, simplifying radicals helps to find the most reduced form of the equation, making it straightforward to solve subsequently.

Isolating Variable

The last task in solving the quadratic equation is to isolate the variable. From our simplified equation \(n + 5 = \pm 4 \sqrt{2}\), we need to get \(n\) alone on one side. To isolate \(n\), subtract 5 from both sides of the equation: \(n = \pm 4 \sqrt{2} - 5\).
Here, we consider both the positive and negative solutions from the ± symbol:

• For the positive: \(n = 4 \sqrt{2} - 5\)
• For the negative: \(n = -4 \sqrt{2} - 5\)

This results in two potential solutions for our original quadratic equation.
Isolating the variable is an essential process that involves performing inverse operations to get the variable by itself, allowing us to find the solutions easily.