What is the value of $\frac{6}{x}+2 x^2$, when $x=3$ ?
The goal is to evaluate the algebraic expression \(\frac{6}{x} + 2x^2\) for a specific given value of \(x = 3\). An algebraic expression is a combination of variables, numbers, and operations. Substituting a numerical value for a variable allows us to determine the overall value of the expression at that point. We will substitute \(x = 3\) into the expression and follow the order of operations to find the result.
Substitute \(x = 3\) into the expression \(\frac{6}{x} + 2x^2\):
\[\frac{6}{3} + 2(3)^2\]
First, we divide 6 by 3:
\[\frac{6}{3} = 2\]
Next, we handle the exponentiation, where \(3^2 = 3 \times 3 = 9\).
Then, we perform the multiplication:
\[2 \times 9 = 18\]
Finally, we add the two results together:
\[2 + 18 = 20\]
By substituting \(x = 3\) into the expression \(\frac{6}{x} + 2x^2\), we found that the value is \(20\). This matches the fourth option provided in the problem. The calculations performed were: \(\frac{6}{3} = 2\), \(3^2 = 9\), and \(2 + (2 \times 9) = 20\). Therefore, the final value of the expression is \(\boxed{20}\).
Evaluating expressions is a fundamental skill in many fields, including finance and engineering. For example, if the cost of producing an item follows the formula \(C(x) = \frac{600}{x} + 2x^2\), where \(x\) is the number of items in hundreds, you could use this calculation to find the cost of production for a specific batch size. By substituting the number of items into the formula, you can efficiently estimate costs, optimize resources, and plan your budget effectively.