Problem 76 - Intermediate Algebra

Question

Multiply and then simplify if possible. $$ (\sqrt{x-6}-7)^{2} $$

Step-by-Step Solution

Answer

The expression simplifies to \(x - 14\sqrt{x-6} + 43\).

Step 1: Apply the Binomial Expansion Formula

The expression \((\sqrt{x-6} - 7)^2\) is a binomial squared. To expand this, we use the binomial expansion formula: \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = \sqrt{x-6}\) and \(b = 7\).

Step 2: Calculate \(a^2\)

Calculate the square of \(a\), which is \((\sqrt{x-6})^2 = x-6\).

Step 3: Calculate \(2ab\)

Next, calculate \(2ab\), which is \(2 \times \sqrt{x-6} \times 7 = 14\sqrt{x-6}\). This is done by multiplying the constants and the radical.

Step 4: Calculate \(b^2\)

Now, calculate \(b^2\) which is \(7^2 = 49\).

Step 5: Combine All Terms

Substitute the expressions from Steps 2, 3, and 4 back into the binomial expansion formula: \( (\sqrt{x-6})^2 - 2 \times \sqrt{x-6} \times 7 + 7^2 = x - 6 - 14\sqrt{x-6} + 49 \).

Step 6: Simplify the Expression

Combine and simplify the constant terms: \( x - 6 + 49 = x + 43 \). Thus, the expanded and simplified expression is \( x - 14\sqrt{x-6} + 43 \).

Key Concepts

Radical Expressions

Radical expressions involve roots, with the square root being one of the most common. In our exercise, we encounter the square root in the term \( \sqrt{x-6} \). The square root symbol \( \sqrt{\cdot} \) indicates that a number is being multiplied by itself to result in the original expression under the root.

• For example, \( \sqrt{9} = 3 \), because \( 3 \times 3 = 9 \).

When working with algebraic problems involving radicals, especially when squared like \((\sqrt{x-6})^2\), the radical "disappear" transforming into the expression \(x-6\).
This happens because squaring a square root cancels them out, simply returning the initial number inside the radical. Understanding these basics can significantly help in solving complex expressions involving radicals.

Simplification of Expressions

Simplifying algebraic expressions is the art of making these expressions more readable and easier to work with. After expanding an expression using binomial identities, it often appears complicated, composed of multiple terms.

• Initially, you might have terms like \(x - 6 - 14\sqrt{x-6} + 49\).
• The task is to combine like terms to reduce the number of terms.

In our solution, we combine the constant terms, \(-6 + 49\), resulting in the simplified form of \(x + 43\).
Simplifying expressions not only helps in making calculations more manageable but also in visualizing the problem better, aiding in understanding the relationships and operations within the expression.

Algebraic Identities

Algebraic identities are potent tools in algebra that allow us to transform and simplify expressions efficiently. An identity is a truth that holds for all variable values, precisely showcasing the reliability in calculations.

• The binomial identity \((a-b)^2 = a^2 - 2ab + b^2\) is a typical example, which we used to expand \((\sqrt{x-6} - 7)^2\).
• Understanding how to apply this identity turns complex algebraic manipulations into straightforward algebraic steps.

By assigning \(a\) and \(b\), finding \(a^2, 2ab,\) and \(b^2\), and recombining them, we reach a simplified form.
This process not only helps in the calculation but deepens one’s algebraic understanding, serving as a fundamental component in both basic and advanced mathematics.