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In mathematics, evaluating a function means to find the value of the function at a specific input. The given function in the exercise is a quadratic function defined as \( f(x) = x^2 \). To evaluate this function for a particular input, you substitute the input value for \( x \) in the function expression.

In this exercise, we substitute \( 7 + \sqrt{3} \) into the function. So, you replace \( x \) with \( 7 + \sqrt{3} \), resulting in \( f(7 + \sqrt{3}) = (7 + \sqrt{3})^2 \). This substitution is the first step in solving the problem.

Understanding how to properly substitute values into a function is crucial for solving various math problems, especially those involving complex expressions.

A binomial expansion involves expanding expressions that are raised to a power. Here, we deal with the square of a binomial: \( (7 + \sqrt{3})^2 \).

The formula for the square of a binomial is: \((a + b)^2 = a^2 + 2ab + b^2\).

This formula helps simplify expressions by breaking them down into more manageable parts. For our problem, \( a = 7 \) and \( b = \sqrt{3} \). Plugging these values into the formula, we get: \( (7 + \sqrt{3})^2 = 7^2 + 2 \cdot 7 \cdot \sqrt{3} + (\sqrt{3})^2 \).

This step by step process makes it easier to see how each part of the expression contributes to the final result. It’s important to follow each step carefully to avoid mistakes.

The final step in solving the problem is simplification, where you combine like terms and perform arithmetic operations to get a cleaner result.

For the expanded binomial \( 7^2 + 2 \cdot 7 \cdot \sqrt{3} + (\sqrt{3})^2 \), we calculate each part separately:

• \( 7^2 = 49 \)
• \( 2 \cdot 7 \cdot \sqrt{3} = 14\sqrt{3} \)
• \((\sqrt{3})^2 = 3 \)

Putting it all together, this leads to: \( 49 + 14\sqrt{3} + 3 \). Combine the like terms (49 and 3) to get: \( 52 + 14\sqrt{3} \).

Simplification helps not only in making the final answer more readable but also ensures accuracy in calculations. When combining like terms, make sure to identify and group similar mathematical expressions.

This step confirms that the final value of \( f(7 + \sqrt{3}) \) is indeed \( 52 + 14\sqrt{3} \).