Math Problem Statement

Expand the expression \(\left(3 \sqrt{x} + \sqrt{2}\right)^2\).

Solution

The expression provided is:

(3x+2)2\left(3 \sqrt{x} + \sqrt{2}\right)^2

To expand this, we can apply the formula for the square of a binomial (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2, where:

  • a=3xa = 3\sqrt{x}
  • b=2b = \sqrt{2}

Step-by-Step Solution:

  1. Square the first term: (3x)2=9x(3 \sqrt{x})^2 = 9x

  2. Multiply the two terms together and double the result: 2×(3x)×(2)=62x2 \times (3 \sqrt{x}) \times (\sqrt{2}) = 6 \sqrt{2x}

  3. Square the second term: (2)2=2(\sqrt{2})^2 = 2

Putting It All Together:

Combining all terms, we get: (3x+2)2=9x+62x+2\left(3 \sqrt{x} + \sqrt{2}\right)^2 = 9x + 6\sqrt{2x} + 2

Would you like further details on any part of this expansion?

Related Questions:

  1. What is the general formula for expanding (a+b)2(a + b)^2?
  2. How do we simplify terms involving square roots?
  3. What happens if x=4x = 4 in this expression?
  4. How can we factor expressions involving square roots?
  5. How would the expression change if we had (3x2)2(3 \sqrt{x} - \sqrt{2})^2?

Tip:

When expanding binomials with square roots, keep each square root in its own term until you can combine terms. This minimizes errors in simplification.

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Math Problem Analysis

Mathematical Concepts

Algebra
Binomial Expansion
Square Roots

Formulas

Square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\)

Theorems

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Suitable Grade Level

Grades 9-10