\( \# 6 \quad x ^ { 2 } - 2 x = 3 \)

To solve the equation \( x^2 - 2x = 3 \), you can start by rearranging it into a standard form. Subtract 3 from both sides to get \( x^2 - 2x - 3 = 0 \). Now, you can factor this quadratic equation as \( (x - 3)(x + 1) = 0 \). Setting each factor to zero gives the possible solutions: \( x = 3 \) and \( x = -1 \). This type of equation appears often in real-world scenarios, like physics and finance, where relationships can be modeled quadratically. For example, predicting the trajectory of a projectile or calculating profit/loss in business after certain variables change can commonly utilize quadratic equations!