19. Which value(s) of \( x \) satisfies the equation below? \[ -8-3 n^{2}=-38 \] a. \( n= \pm 10 \) b. \( n= \pm 5 \) c. \( n= \pm \sqrt{5} \) d. \( n= \pm \sqrt{10} \)

To solve the equation \(-8 - 3n^{2} = -38\), first isolate the term involving \(n^{2}\). Adding 8 to both sides gives: \(-3n^{2} = -30\) Next, divide both sides by -3: \(n^{2} = 10\) Now, taking the square root of both sides results in: \(n = \pm \sqrt{10}\) Thus, the correct answer is \(n = \pm \sqrt{10}\), corresponding to option (d). Also, it's interesting to note that equations like this show up frequently in physics problems, especially when dealing with projectile motion or energy conservation, where quadratic terms represent kinetic or potential energy. Understanding how to manipulate these equations is crucial for both math and science students! Many students mistakenly forget to include both the positive and negative roots when taking the square root of a number. This can lead to incomplete solutions, so always remember that if \(n^2 = a\), then \(n\) can be both \(\sqrt{a}\) and \(-\sqrt{a}\)!