Let's simplify the expression \((\sqrt{x-1} \cdot (6x + 4)) \cdot 2\sqrt{x-1}\).

### Step 1: Simplify the First Part

First, simplify \(\sqrt{x-1} \cdot (6x + 4)\):

\[
\sqrt{x-1} \cdot (6x + 4) = 6x\sqrt{x-1} + 4\sqrt{x-1}
\]

### Step 2: Multiply by \(2\sqrt{x-1}\)

Now, multiply the entire expression by \(2\sqrt{x-1}\):

\[
\left(6x\sqrt{x-1} + 4\sqrt{x-1}\right) \cdot 2\sqrt{x-1}
\]

Distribute \(2\sqrt{x-1}\) to each term:

1. \(6x\sqrt{x-1} \cdot 2\sqrt{x-1} = 12x(\sqrt{x-1})^2\)
2. \(4\sqrt{x-1} \cdot 2\sqrt{x-1} = 8(\sqrt{x-1})^2\)

### Step 3: Simplify the Squared Terms

Since \((\sqrt{x-1})^2 = x - 1\), replace \(\sqrt{x-1}\) with \(x-1\):

\[
12x(x-1) + 8(x-1)
\]

### Step 4: Distribute and Combine

Now, distribute the terms:

\[
12x(x - 1) = 12x^2 - 12x
\]
\[
8(x - 1) = 8x - 8
\]

Combine all the terms:

\[
12x^2 - 12x + 8x - 8 = 12x^2 - 4x - 8
\]

### Final Answer

The simplified expression is:

\[
12x^2 - 4x - 8
\]

Would you like more details or have any questions? 

Here are five related questions:

1. How do you factor quadratic expressions like \(12x^2 - 4x - 8\)?
2. What are the steps to simplify expressions involving square roots?
3. How can the distributive property be used in algebraic expressions?
4. How do you combine like terms in polynomial expressions?
5. How does multiplying binomials relate to the FOIL method?

**Tip:** Always simplify square roots early when possible to reduce complexity in algebraic expressions.

Learn how to simplify the expression sqrt(x-1) * (6x + 4) * 2sqrt(x-1) with a detailed step-by-step solution. This problem covers algebraic concepts involving simplification and multiplication of square roots. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation