Problem 34 - Intermediate Algebra

Question

Use Property \(6.1\) to help solve each quadratic equation. $$ n^{2}-54=0 $$

Step-by-Step Solution

Answer

The solutions are \(n = 3\sqrt{6}\) and \(n = -3\sqrt{6}\).

Step 1: Identify the Structure

The given quadratic equation is in the form of \(n^2 - 54 = 0\). This is a simple equation since it does not contain a coefficient for the linear term \(n\).

Step 2: Use Property for Solving Quadratics

Property 6.1 likely refers to the quadratic equation property where if \(n^2 = a\), then \(n = \pm\sqrt{a}\). First, add \(54\) to both sides to isolate \(n^2\): \(n^2 = 54\).

Step 3: Solve for n using Square Roots

Apply the square root property. Since \(n^2 = 54\), then \(n = \pm\sqrt{54}\). Simplify the square root: \(n = \pm\sqrt{54} = \pm\sqrt{9 \times 6} = \pm 3\sqrt{6}\).

Key Concepts

Square Roots

Square roots are a foundational concept in mathematics, especially in algebra. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because \( 4 imes 4 = 16 \).
When dealing with equations such as the quadratic equation \( n^2 = 54 \), finding the square root helps isolate the variable, which leads us to the solution. The principle of square roots (Property 6.1 in our exercise) tells us if \( n^2 = a \), then \( n = \pm \sqrt{a} \). This means the solution can either be positive or negative.
It's important to remember:

• Not all numbers have a neat square root. Some are irrational, like \( \sqrt{54} \).
• Solving by square roots is straightforward for equations that are already set equal to a constant, as \( n^2 = \text{constant} \).

Simplifying Radicals

Simplifying radicals is essential for expressing square roots in their simplest form. For instance, \( \sqrt{54} \) can be simplified by factoring the number under the radical sign. The goal is to find perfect squares.
Here's a way to simplify:

• Find the largest perfect square dividing the radical number. Here, \( 9 \) is a perfect square that divides 54.
• Express the number under the radical as a product of the perfect square and another factor: \( \sqrt{54} = \sqrt{9 \times 6} \).
• Take the square root of the perfect square out of the radical: \( \sqrt{9} = 3 \), so \( \sqrt{54} = 3\sqrt{6} \).

Simplifying helps in making calculations easier to handle and understand in math and science problems, keeping results concise and clear.

Solving Equations

Solving equations involves finding the value of unknown variables that make the equation true. Quadratic equations are equations where the highest power of the variable is a square.
The equation in our example is \( n^2 - 54 = 0 \). To solve it:

• First, isolate \( n^2 \) by adding 54 to both sides to get \( n^2 = 54 \).
• Use the square root method, which means taking the square root of both sides: \( n = \pm \sqrt{54} \).
• This results in two possible solutions because of the \( \pm \), representing both the positive and negative roots \( n = \pm 3\sqrt{6} \).

Solving quadratic equations often require different techniques, such as factoring, completing the square, or using the quadratic formula. Choosing the right approach depends on the structure of the equation.