Respuesta :
Solution:
Consider the following equation:
[tex]x^4-6x^2-7x-6=0[/tex]First, to solve this equation, we must factor it:
[tex](x+2)(x-3)(x^2+x+1)=0[/tex]using the zero factor theorem, the following must be met:
Equation 1:
[tex](x+2)=0[/tex]or
Equation 2:
[tex](x-3)=0[/tex]or
Equation 3:
[tex](x^2+x+1)=0[/tex]From equation 1, solving for x, we obtain:
[tex]x\text{ = -2}[/tex]or from equation 2, solving for x, we obtain:
[tex]x\text{ = 3}[/tex]Now, remember the following quadratic formula to find the solutions of a quadratic equation:
[tex]\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]applying this to equation 3, where
a = 1
b= 1
and
c = 1
we obtain:
[tex]\frac{-1\pm\sqrt[]{1-4}}{2}=\frac{-1\pm\sqrt[]{-3}}{2}=\frac{-1\pm\sqrt[]{3}i\text{ }}{2}=-\frac{1}{2}\pm\frac{\sqrt[]{3}i\text{ }}{2}[/tex]then, we obtain two additional solutions:
[tex]-\frac{1}{2}+\frac{\sqrt[]{3}i\text{ }}{2}[/tex]or
[tex]-\frac{1}{2}-\frac{\sqrt[]{3}i\text{ }}{2}[/tex]so that, we can conclude that the correct answer is:
The solutions (zeros) for the given equation are:
[tex]x\text{ = -2}[/tex][tex]x\text{ = 3}[/tex][tex]-\frac{1}{2}+\frac{\sqrt[]{3}i\text{ }}{2}[/tex][tex]-\frac{1}{2}-\frac{\sqrt[]{3}i\text{ }}{2}[/tex]