## What does it mean to solve over a set of complex numbers?

Over the complex numbers, every polynomial (with real-valued coefficients) can be factored into a product of linear factors. We can state this also in root language: Over the complex numbers, every polynomial of degree n (with real-valued coefficients) has n roots, counted according to their multiplicity.

## What is the set of complex numbers?

A complex number is a number that can be written in the form a + b i a + bi a+bi, where a and b are real numbers and i is the imaginary unit defined by i 2 = − 1 i^2 = -1 i2=−1. The set of complex numbers, denoted by C, includes the set of real numbers (R) and the set of pure imaginary numbers.

**How do you find the real solution?**

If the value of the discriminant is positive, there are two real solutions for x, meaning the graph of the solution has two distinct x-intercepts. If the value of the discriminant is zero, there is one real solution for x, meaning the graph of the solution has one x-intercept.

**Are complex solutions real solutions?**

The expression b2 − 4ac is called the discriminant, and can be used to determine whether the solutions are real, repeated, or complex: 1) If the discriminant is less than zero, the equation has two complex solution(s). 2) If the discriminant is equal to zero, the equation has one repeated real solution(s).

### What is 3i equal to?

Remember that a complex number has the form a + bi. You need to figure out what a and b need to be. Since −3i is an imaginary number, it is the imaginary part (bi) of the complex number a + bi. This imaginary number has no real parts, so the value of a is 0….

Imaginary Numbers | |
---|---|

3i (b = 3) | −672i (b = −672) |

(b = ) | (b = ) |

### What are real and complex solutions?

If the discriminant equals 0, then the equation has one real solution, a double root. If the discriminant is less than 0, then the equation has two complex solutions. If the discriminant is greater than 0, then the equation has two real solutions.