Verified Answer :

  • Recognize the binomial square identity \((a+b)^2 = a^2 + 2ab + b^2\).
  • Substitute \(a=3\) and \(b=\sqrt{2}\) into the identity.
  • Expand and simplify the terms to obtain \(9 + 6\sqrt{2} + 2\).
  • The final simplified result is \(\boxed{11+6\sqrt{2}}\).

Explanation

  1. Identifying the Algebraic Structure

The goal is to simplify the expression \((3+\sqrt{2})^2\). This expression is a classic example of a binomial square, which can be expanded using the algebraic identity:
\[(a+b)^2 = a^2 + 2ab + b^2\]
In our case, we identify the terms as \(a = 3\) and \(b = \sqrt{2}\). By applying this identity, we can break the expression into manageable components.

  1. Calculating the Expanded Terms

Now we calculate each component of the expansion individually:

  1. \(a^2 = 3^2 = 9\)
  2. \(2ab = 2 \times 3 \times \sqrt{2} = 6\sqrt{2}\)
  3. \(b^2 = (\sqrt{2})^2 = 2\)

By substituting these values back into the formula \(a^2 + 2ab + b^2\), we get:
\[(3+\sqrt{2})^2 = 9 + 6\sqrt{2} + 2\]

  1. Final Simplification

We combine the constant terms to simplify the expression:
\[9 + 2 + 6\sqrt{2} = 11 + 6\sqrt{2}\]
Thus, the simplified form of \((3+\sqrt{2})^2\) is \(11+6\sqrt{2}\), which corresponds to the first option provided.

Examples

The identity \((a+b)^2 = a^2+2ab+b^2\) is incredibly useful in geometry and engineering. For instance, if you are designing a square floor area that is extended on two sides by \(\sqrt{2}\) meters, the new area is exactly \((3+\sqrt{2})^2\). Instead of multiplying decimals, applying this formula allows you to quickly calculate the exact area needed for materials, ensuring precision in your construction plans. Understanding these expansions helps in simplifying complex physical measurements in real-world engineering.

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