he foci of the ellipse defined by the equation \( \frac{(x-5)^{2}}{25}+\frac{(y+1)}{4} \) essary, round to the nearest tenth.

To find the foci of the ellipse given by the equation \( \frac{(x-5)^{2}}{25}+\frac{(y+1)^{2}}{4} = 1 \), we first identify the center, which is at the point \((5, -1)\). The semi-major axis length \(a\) is \(5\) (the square root of \(25\)), and the semi-minor axis length \(b\) is \(2\) (the square root of \(4\)). The distance \(c\) from the center to each focus can be calculated using the formula \(c = \sqrt{a^{2} - b^{2}}\). Here, \(c = \sqrt{25 - 4} = \sqrt{21} \approx 4.6\). Therefore, the foci are located at \((5 \pm c, -1)\), which rounds to approximately \((5 \pm 4.6, -1)\) or \((9.6, -1)\) and \((0.4, -1)\). Now, let's have some fun with this! Elipses aren't just geometric shapes; they're like cosmic highways! Imagine planets orbiting a sun (the center), with their distance to that sun changing as they travel—those paths can be described by ellipses. Ellipses are everywhere; from the paths of satellites to the way that sound waves travel in concert halls, they ensure that we experience everything from music to satellite communication in a harmonious way. If you ever find yourself miscalculating, fear not! A common mistake is to confuse which axis is longer; always check which denominator is larger to ensure you're identifying the correct semi-major and semi-minor axes. Another tip is to remember the standard form of the ellipse equation: keep your terms organized and check that everything is set to equal 1. Happy calculating!