Problem 46 - Elementary Algebra

Question

Solve using the Square Root Property. \((n-7)^{2}-8=64\)

Step-by-Step Solution

Answer

n = 7 \pm 6\sqrt{2}

Step 1 - Isolate the squared term

Start by isolating the squared term \( (n-7)^{2} \). Add 8 to both sides of the equation: \( (n-7)^{2} = 64 + 8 \). This simplifies to \( (n-7)^{2} = 72 \).

Step 2 - Apply the Square Root Property

Use the Square Root Property, which says if \( x^{2} = k \), then \( x = \pm\sqrt{k} \). Apply this to both sides: \( n-7 = \pm\sqrt{72} \).

Step 3 - Simplify the square root

Simplify \( \sqrt{72} \). Notice \( \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2} \). So \( n-7 = \pm 6\sqrt{2} \).

Step 4 - Solve for \( n \)

Solve for \( n \) by adding 7 to both sides. This gives two solutions: \( n = 7 + 6\sqrt{2} \) and \( n = 7 - 6\sqrt{2} \).

Key Concepts

Solving Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest power of the variable is squared.
These equations often have the form \(ax^2 + bx + c = 0\).
To solve them, we can use several methods such as factoring, completing the square, and the quadratic formula.
This article will focus on solving quadratic equations using the Square Root Property.
This method is straightforward but requires isolating the squared term first.
Once isolated, you can take the square root of both sides to find the variable.

Isolating the Squared Term

To utilize the Square Root Property, we first need to isolate the squared term.
Given the equation \((n-7)^2 - 8 = 64\), we begin by adding 8 to both sides:
\[(n-7)^2 = 64 + 8\]
Simplify the right-hand side to get:
\[(n-7)^2 = 72\]
This step is crucial because it sets up the equation to take the square root of both sides, bringing us closer to the solution.

Simplifying Square Roots

The next step is to apply the Square Root Property. If we have an equation of the form \(x^2 = k\), we solve it by taking the square root of both sides to get:
\[x = \pm \sqrt{k}\]
Applying this to our equation:
\[n-7 = \pm \sqrt{72}\]
Simplify the square root \(\sqrt{72}\). Notice \(\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}\).
Therefore, we have:
\[n-7 = \pm 6\sqrt{2}\]

Positive and Negative Roots

The last step involves solving for \(n\). We've simplified our equation to:
\[n-7 = 6\sqrt{2}\] and \[n-7 = -6\sqrt{2}\]
Since we have two equations, we add 7 to both sides to isolate \(n\):
\[n = 7 + 6\sqrt{2}\]
and \[n = 7 - 6\sqrt{2}\]
These are our two solutions:

• \(n = 7 + 6\sqrt{2}\)
• \(n = 7 - 6\sqrt{2}\)

Positive and negative roots result from the nature of square roots. Every positive number has two square roots: the positive root and the negative root.