Answer:
The equation is [tex]x^{2} +2\cdot x + 2 = 0[/tex]-
Step-by-step explanation:
According to the statement, we have a second order polynomial of the form [tex]A\cdot x^{2}+ B\cdot x + C = 0[/tex], whose solutions can be found by Quadratic Formula:
[tex]x_{1,2} = \frac{-B\pm \sqrt{B^{2}-4\cdot A\cdot C}}{2\cdot A}[/tex] (1)
Where [tex]A[/tex], [tex]B[/tex] and [tex]C[/tex] are the coefficients of the polynomial.
If roots are conjugated complex numbers, then:
[tex]B^{2}-4\cdot A\cdot C < 0[/tex] (2)
[tex]B^{2} < 4\cdot A \cdot C[/tex]
If we know that [tex]A = 1[/tex], [tex]C = 2[/tex] and [tex]x_{1,2} = -1\pm i[/tex], then we find that:
[tex]-1\pm i = -\frac{B}{2}\pm \frac{\sqrt{B^{2}-8}}{2}[/tex]
[tex]-2 \pm i\,2 = -B \pm \sqrt{B^{2}-8}[/tex]
By comparing each side, we have the following system of equations:
[tex]-B = -2[/tex] (3)
[tex]\sqrt{B^{2}-8} = 2[/tex] (4)
Whose solution is [tex]B = 2[/tex].
In a nutshell, the equation is [tex]x^{2} +2\cdot x + 2 = 0[/tex]-
