Solve for x: -1+1/x+x+x^2+...+x^n+...=10/3, where |x|<1

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Ответ:

dangolstein03

24.08.2021

Mathematics

$x=\pm\sqrt{2}i\text{ (or) }x=\pm i$

Step-by-step explanation:

We have been given an equation x^4 + 3x^2 + 2 = 0 . We are asked to find the solutions of our given equation using u-substitution.

We can rewrite our given equation as:

(x^2)^2+3x^2+2 = 0

Let us assume that u=x^2 .

u^2+3u+2 = 0

u^2+2u+u+2 = 0

u(u+2)+1(u+2) = 0

(u+2)(u+1) = 0

$(u+2)=0\text{ (or) }(u+1) = 0$

$u=-2\text{ (or) }u=-1$

Upon substituting back the value of u, we will get:

$x^2=-2\text{ (or) }x^2=-1$

$x=\pm\sqrt{-2}\text{ (or) }x=\pm\sqrt{-1}$

$x=\pm\sqrt{-1\cdot 2}\text{ (or) }x=\pm\sqrt{-1}$

Now we will use imaginary unit i. We know that i^2=-1 .

$x=\pm\sqrt{i^2\cdot 2}\text{ (or) }x=\pm\sqrt{i^2}$

$x=\pm\sqrt{2}i\text{ (or) }x=\pm i$

Therefore, the solutions for our given equation are $x=\pm\sqrt{2}i\text{ (or) }x=\pm i$ .

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