Answer:
The correct answer is option
[tex]d.\ x=-i, i, 3 \sqrt{5}\ or\ -3 \sqrt{5}[/tex]
Step-by-step explanation:
The given equation has a degree 4 (Highest power of [tex]x[/tex]):
[tex]x^4-44x^2-45=0[/tex] is the given equation which can be written as:
[tex](x^2)^2-44x^2-45=0 ...... (1)[/tex]
Let [tex]t=x^{2}[/tex] and putting it in equation (1):
[tex]t^2-44t-45=0 \\[/tex]
Solving the above quadratic equation in variable [tex]t[/tex]:
[tex]\Rightarrow t^2-45t+t-45=0 \\\Rightarrow t(t-45)+1(t-45)=0\\\Rightarrow (t+1)(t-45)=0\\\Rightarrow t =-1\ or\ 45[/tex]
We know that [tex]t=x^{2}[/tex]
So, [tex]x^{2} =-1\ or\ x^{2}= 45[/tex]
1. Solving [tex]x^{2} =-1[/tex]
[tex]\Rightarrow x = +\sqrt{-1}\ or\ -\sqrt{-1}\\\Rightarrow x = i\ or\ -i[/tex]
2. Solving [tex]x^{2} =45[/tex]
[tex]\Rightarrow x = +\sqrt{45}\ or\ -\sqrt{45}\\\Rightarrow x = 3\sqrt{5}\ or\ -3\sqrt{5}[/tex]
Hence, correct answer is:
[tex]x=-i, i, 3 \sqrt{5}\ or\ -3 \sqrt{5}[/tex]
