What is the simplest form of the expression $2 x(x-6)-7 x^2-(13 x-3)$?
We are tasked with simplifying the algebraic expression \(2x(x - 6) - 7x^2 - (13x - 3)\). This expression involves multiplying a term into a binomial, subtracting a quadratic term, and subtracting a polynomial by applying the distributive property of multiplication over addition (or subtraction). We will perform this step-by-step to arrive at the simplest form.
First, we distribute the \(2x\) across the terms inside the parentheses:
\[2x(x - 6) = 2x \cdot x - 2x \cdot 6 = 2x^2 - 12x.\]
Next, we distribute the negative sign to the terms inside the second set of parentheses:
\[-(13x - 3) = -13x + 3.\]
Now, we substitute these back into the original expression:
\[(2x^2 - 12x) - 7x^2 - 13x + 3.\]
Now we group the like terms together based on their powers of \(x\):
Combining these results, we get the simplified expression:
\[-5x^2 - 25x + 3.\]
Comparing our result \(-5x^2 - 25x + 3\) with the given options, we find it corresponds exactly to the second choice. Therefore, the simplest form of the expression is \(-5x^2 - 25x + 3\). Thus, the final answer is \(\boxed{-5x^2 - 25x + 3}\).
Simplifying algebraic expressions is a foundational skill in fields like engineering and economics. For instance, if you are modeling the cost of producing a certain number of units, you might end up with an equation representing total costs. By simplifying these expressions, you can quickly identify variable and fixed costs, making it easier to calculate your break-even point or optimize your production strategy. Being able to combine like terms efficiently helps in turning complex models into manageable, actionable insights.