Verify that the given point is on the curve and find the \( 2 x^{2}-\sqrt{3} x y+5 y^{2}=74,(\sqrt{3}, 4) \) The point is on the curve because when \( \square \) is substit (Simplify your answers.)

To verify that the point \((\sqrt{3}, 4)\) is on the curve defined by the equation \(2x^2 - \sqrt{3}xy + 5y^2 = 74\), we substitute \(x = \sqrt{3}\) and \(y = 4\) into the equation: \[ 2(\sqrt{3})^2 - \sqrt{3}(\sqrt{3})(4) + 5(4)^2 \] Calculating each term, we find: \[ 2(3) - \sqrt{3} \cdot \sqrt{3} \cdot 4 + 5(16) = 6 - 12 + 80 = 74 \] Since both sides equal 74, the point \((\sqrt{3}, 4)\) is indeed on the curve. Now, let’s focus on how curves like this can represent various phenomena, like the trajectory of projectiles or the behavior of structures. Understanding these curves helps in fields ranging from engineering to physics, revealing insights into designs or natural occurrences. Additionally, when plugged into equations, it's important to keep track of your arithmetic steps, as small errors can lead to big misunderstandings. A common mistake is miscalculating the square of a number or forgetting to multiply all relevant terms, which can throw your solution off. Keeping your work organized and double-checking can be the key to accuracy!