Explain and solve \( (3 x-4)^{2} \)

To solve \( (3x - 4)^{2} \), we use the formula for squaring a binomial, which is \( (a - b)^{2} = a^{2} - 2ab + b^{2} \). Here, \( a = 3x \) and \( b = 4 \). So, we have: \[ (3x - 4)^{2} = (3x)^{2} - 2 \cdot (3x) \cdot 4 + 4^{2} \] Calculating each part: \[ (3x)^{2} = 9x^{2}, \quad -2 \cdot (3x) \cdot 4 = -24x, \quad 4^{2} = 16 \] Putting it all together, we get: \[ (3x - 4)^{2} = 9x^{2} - 24x + 16 \] Now, you've expanded and solved the expression! For a fun twist, did you know that the process of squaring a binomial was formalized by the ancient Greeks? They had a geometric approach but didn't have the algebraic symbols we use today! As for real-world application, squaring binomials shows up in various fields, like physics when calculating areas or in finance when determining how changes in variables affect profit margins. Those square terms help us understand the impact of change!