Problem 76 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt{x-6}-7)^{2} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Expression

The given expression is \((\sqrt{x-6} - 7)^2\). This is a square of a binomial, which means we can use the formula for the square of a binomial: \((a-b)^2 = a^2 - 2ab + b^2\).

Step 2: Apply the Formula

To expand the expression \((\sqrt{x-6} - 7)^2\), we identify \(a = \sqrt{x-6}\) and \(b = 7\). So according to the formula \((a-b)^2 = a^2 - 2ab + b^2\), we have: \( (\sqrt{x-6})^2 - 2(\sqrt{x-6})(7) + 7^2 \).

Step 3: Calculate Each Term

First, calculate \((\sqrt{x-6})^2\), which simplifies to \(x-6\). Next, calculate the middle term: \(-2 \times \sqrt{x-6} \times 7 = -14\sqrt{x-6}\).Finally, calculate \(7^2 = 49\).

Step 4: Combine the Terms

Substitute and combine all the calculated terms into the expression: \(x-6 - 14\sqrt{x-6} + 49\).

Step 5: Simplify the Expression

Combine the constant terms: \(x - 6 + 49 = x + 43\).Thus, the expression simplifies to: \(x + 43 - 14\sqrt{x-6}\).

Key Concepts

Binomial Expansion

When we encounter an expression like \((\sqrt{x-6} - 7)^2\), we are dealing with a binomial expansion. A binomial is an expression that contains two distinct terms. In this case, the two terms are \(\sqrt{x-6}\) and \(-7\). Using the binomial expansion, we apply the formula for the square of a binomial: \((a-b)^2 = a^2 - 2ab + b^2\). This formula is extremely handy because it allows us to expand complex expressions into simpler components.

Here's the step-by-step approach:

• Identify the terms: Here, \(a\) is \(\sqrt{x-6}\) and \(b\) is \(7\).
• Apply the formula: Substitute these into the formula to get \((\sqrt{x-6})^2 - 2\times\sqrt{x-6}\times 7 + 7^2\).
• Calculate each term separately.

By using this systematic approach, the binomial expansion becomes much more approachable.

Simplifying Expressions

Once we have used the binomial expansion, it's essential to simplify the expression. Simplifying helps in reducing the expression to its simplest form, making it easier to interpret and solve.

After applying the formula, we have the terms: \(x-6\), \(-14\sqrt{x-6}\), and \(49\). To simplify, we need to combine like terms:

• Combine the constants: \(-6 + 49 = 43\).
• Place this along with the other term, \(-14\sqrt{x-6}\).

This gives us the simplified expression \(x + 43 - 14\sqrt{x-6}\). Simplifying expressions is a microscopic but crucial step. It ensures that the expression is in its most readable and simplified version.

Square Roots

Square roots can initially seem complex, but they're simply a value that, when multiplied by itself, gives the original number. In our problem, we deal with the square root expression \(\sqrt{x-6}\).

When squaring a square root, such as \((\sqrt{x-6})^2\), the result is straightforward: it eliminates the square root, giving \(x-6\). This is because the square and the square root are inverse operations.

• In any operation, squaring a square root returns the number under the root sign.
• Understanding this concept helps in simplifying terms like \(\sqrt{x-6}\) when they appear within larger expressions.

Understanding how to manipulate square roots efficiently simplifies many algebraic problems, making them more approachable.