Question Video: Solving Quadratic Equations over the Set of Complex Numbers Mathematics

Video Transcript

Determine the solution set of four 𝑥 squared plus 111 equals 75 over the set of complex numbers.

We can begin to solve the quadratic equation by subtracting 111 from both sides. 75 minus 111 is equal to negative 36. So we have four 𝑥 squared is equal to negative 36. We can then divide both sides of this equation by four. This gives us 𝑥 squared is equal to negative nine. Square rooting both sides of this equation, we have 𝑥 is equal to the positive or negative of the square root of negative nine.

We recall from our knowledge of complex and imaginary numbers that the square root of negative one is equal to 𝑖. Using our laws of radicals or surds, the square root of negative nine can be rewritten as the square root of nine multiplied by the square root of negative one. As the square root of nine is equal to three, the square root of negative nine is equal to three 𝑖. 𝑥 is therefore equal to positive or negative three 𝑖.

The solution set of the quadratic equation four 𝑥 squared plus 111 equals 75 has two elements: negative three 𝑖 and three 𝑖.