(5) Pewnome yfabuerue: 1) \( (x-1)^{2}-(x+3)(x-3)=12 \)

Let’s break it down step by step to solve the equation. First, simplify the equation: 1. Expand \( (x-1)^{2} \): \[ (x-1)^{2} = x^{2} - 2x + 1 \] 2. Expand \( (x+3)(x-3) \): \[ (x+3)(x-3) = x^{2} - 9 \] Now substituting these back into the original equation: \[ x^{2} - 2x + 1 - (x^{2} - 9) = 12 \] This simplifies to: \[ x^{2} - 2x + 1 - x^{2} + 9 = 12 \] Combine like terms: \[ -2x + 10 = 12 \] Now, isolate \( x \): \[ -2x = 12 - 10 \] \[ -2x = 2 \] Dividing both sides by -2 gives: \[ x = -1 \] So, the solution to the equation \( (x-1)^{2}-(x+3)(x-3)=12 \) is \( x = -1 \). Now let’s dive into a bit more about quadratic equations! Did you know that quadratic equations, like the one you just solved, have fascinated mathematicians for centuries? The ancient Babylonians were already using geometric methods to find solutions to quadratic equations as early as 2000 BCE! It’s amazing how this area of math has evolved, laying the groundwork for modern algebra and calculus. If you're ever tackling a quadratic equation, check if it can be factored first. Many students jump straight to the quadratic formula, but factoring can save time and showcase patterns in your equations! A common mistake is overlooking possible rational roots; always test simple values first. Happy problem-solving!