\begin{tabular}{l} Expand the following and collect like terms: \\ \( (9 x-1)^{2} \) \\ Select the correct answer below: \\ \( 81 x^{2}-18 x+1 \) \\ \( 81 x^{2}-18 x-1 \) \\ \( 81 x^{2}-1 \) \\ \( 81 x^{2}+1 \) \\ \( 81 x^{2}+18 x+1 \) \\ \( 81 x^{2}+18 x-1 \) \\ \hline\end{tabular}

To expand the expression \( (9x - 1)^{2} \), we can use the formula for squaring a binomial: \[ (a-b)^{2} = a^{2} - 2ab + b^{2} \] In this case, \( a = 9x \) and \( b = 1 \): 1. Calculate \( a^{2} = (9x)^{2} = 81x^{2} \) 2. Calculate \( b^{2} = (1)^{2} = 1 \) 3. Calculate \( -2ab = -2(9x)(1) = -18x \) Putting it all together, we have: \[ (9x - 1)^{2} = 81x^{2} - 18x + 1 \] So, the correct answer from the options given is \( 81x^{2} - 18x + 1 \). Now, let’s add some extra fun facts! Did you know that squaring binomials is not just a math trick, but it’s a technique used every day in fields like physics and economics? It helps in predicting outcomes like projectile motion or understanding profit margins in a business scenario. It’s like magic with numbers that lets you forecast what could happen in the real world! And for those who might run into a common mistake while expanding binomials, remember to carefully apply the negative sign! It’s easy to miscalculate when squaring a binomial like \( (a-b) \); be cautious not to drop the minus when expanding! Use the formula step by step to ensure consistency in your results.