Find the solution set of 𝑥 squared plus eight 𝑥 plus 185 equals zero, given that 𝑥 is in the complex numbers.
We can use the quadratic formula to solve for the solution set. For an equation 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, we can take negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 all divided by two 𝑎. So for our equation 𝑥 squared plus eight 𝑥 plus 185, 𝑎 is one, 𝑏 is eight, and 𝑐 is 185. So we plug in one for 𝑎, eight for 𝑏, and 185 for 𝑐.
Now we have negative eight plus or minus the square root of 64 minus 740 divided by two. 64 minus 740 is negative 676. And since we have a negative underneath the square root, we’re gonna have an imaginary number. So we’re gonna have an 𝑖. So we have negative eight plus or minus 26𝑖 divided by two.
Since negative eight and 26 are both divisible by two, we get negative four plus or minus 13𝑖. So our final answer is negative four plus 13𝑖 and negative four minus 13𝑖.