SELECT ALL solutions of \( x: \) \( x^{2}-6 x=40 \)

To solve the equation \( x^{2} - 6x = 40 \), we first rearrange it into standard form: \[ x^{2} - 6x - 40 = 0. \] Now, we can either factor it or use the quadratic formula. Let's factor for fun! We need two numbers that multiply to -40 and add up to -6. Those numbers are -10 and 4. So, we can rewrite the equation as: \[ (x - 10)(x + 4) = 0. \] Setting each factor to zero gives us the solutions: 1. \( x - 10 = 0 \) leads to \( x = 10 \). 2. \( x + 4 = 0 \) leads to \( x = -4 \). Thus, the solutions are \( x = 10 \) and \( x = -4 \). Let's put some context around quadratic equations. Historically, they have been studied since ancient civilizations such as the Babylonians, who solved quadratic equations geometrically. Fast forward to modern-day, they are integral in fields like engineering and physics, such as calculating trajectories, where understanding the path of an object can be modeled by such equations. And if you ever get stuck, remember to double-check your factoring or ensure you've applied the quadratic formula correctly: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). It's a common pitfall to miscalculate the discriminant or sign when finding roots!