Answer:
The fundamental theorem of algebra guarantees that a polynomial equation has the same number of complex roots as its degree.
Step-by-step explanation:
The fundamental theorem of algebra guarantees that a polynomial equation has the same number of complex roots as its degree.
We have to find the roots of this given equation.
If a quadratic equation is of the form [tex]ax^{2}+bx +c=0[/tex]
Its roots are [tex]\frac{-b+\sqrt{b^{2}-4ac } }{2a}[/tex] and [tex]\frac{-b-\sqrt{b^{2}-4ac } }{2a}[/tex]
Here the given equation is [tex]2x^{2}-4x-1[/tex] = 0
a = 2
b = -4
c = -1
If the roots are [tex]x_{1} and x_{2}[/tex], then
[tex]x_{1}[/tex] = [tex]\frac{-2+\sqrt{(-4)^{2}-4\times 2\times (-1)}}{2\times 2}[/tex]
= [tex]\frac{4 +\sqrt{24}}{4}[/tex]
= [tex]\frac{2+\sqrt{6} }{2}[/tex]
[tex]x_{2}[/tex] = [tex]\frac{-2-\sqrt{(-4)^{2}-4\times 2\times (-1)}}{2\times 2}[/tex]
= [tex]\frac{4 +\sqrt{8}}{4}[/tex]
= [tex]\frac{2-\sqrt{6} }{2}[/tex]
These are the two roots of the equation.
