Question

What does the fundamental theorem of algebra illustrate?

Answer 1

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 $ax^{2}+bx +c=0$

Its roots are $\frac{-b+\sqrt{b^{2}-4ac } }{2a}$ and $\frac{-b-\sqrt{b^{2}-4ac } }{2a}$

Here the given equation is $2x^{2}-4x-1$ = 0

a = 2

b = -4

c = -1

If the roots are $x_{1} and x_{2}$, then

$x_{1}$ = $\frac{-2+\sqrt{(-4)^{2}-4\times 2\times (-1)}}{2\times 2}$

                       = $\frac{4 +\sqrt{24}}{4}$

                       = $\frac{2+\sqrt{6} }{2}$

$x_{2}$ = $\frac{-2-\sqrt{(-4)^{2}-4\times 2\times (-1)}}{2\times 2}$

                        = $\frac{4 +\sqrt{8}}{4}$

                        = $\frac{2-\sqrt{6} }{2}$

These are the two roots of the equation.