Answer:
[tex]\left \{ {{x_{1} =+\sqrt{7} } \atop {x_{2} =-\sqrt{7}}} \right.[/tex]
[tex]\left \{ {{x_{3} =+\sqrt{2}i } \atop {x_{4} =-\sqrt{2}i}} \right.[/tex]
Step-by-step explanation:
The given equation is
[tex]x^{4}-5x^{2} -14=0[/tex]
To factor this expression, we can change variables to make easier to see the solution.
[tex]x^{4}=y^{2}[/tex] and [tex]x^{2} = y[/tex]
Applying the change, we have
[tex]y^{2}-5y-14=0[/tex]
Now, you can observe that to factor this expression, we just need to find two number which product is 14 and which difference is 5, such numbers are 7 and 2.
[tex]y^{2}-5y-14=(y-7)(y+2)[/tex]
However, remember that [tex]y=x^{2}[/tex], so
[tex]x^{4}-5x^{2} -14=(x^{2}-7 )(x^{2}+2)=0[/tex]
Applying the zero property, we have
[tex]x^{2} -7=0 \implies x^{2} =7 \implies \left \{ {{x_{1} =+\sqrt{7} } \atop {x_{2} =-\sqrt{7}}} \right.[/tex]
[tex]x^{2} +2=0 \implies x^{2} =-2 \implies \left \{ {{x_{3} =+\sqrt{2}i } \atop {x_{4} =-\sqrt{2}i}} \right.[/tex]
As you can see, the equation has two real solutions and two complex solutions.
Therefore, the right answer is the last choice.
