Question

(x^2+1)/(x^2-1)+6(x^2-1)/(x^2+1)-5=0 решите с введением новой переменной

Answer 1

$\(\frac{x^2+1}{x^2-1}+6\frac{x^2-1}{x^2+1}-5=0\)$

ОДЗ: х≠±1

Замена: $\(\frac{x^2+1}{x^2-1}=t\)$, t≠0

$\(t+6\frac{1}{t}-5=0\)$

$\(t^2+6-5t=0\)$

$\(t^2-5t+6=0\)$

$\(t_1=2;\,t_2=3\)$

Возвращаемся к замене:

$\(\frac{x^2+1}{x^2-1}=2\)$ или  $\(\frac{x^2+1}{x^2-1}=3\)$

1) $\(x^2+1=2(x^2-1);\,\)$ $\(x^2+1=2x^2-2;\,\)$

$\(x^2=3;\,\)$ $\(x_1=\sqrt{3};\,x_2=-\sqrt{3}.\)$

2) $\(x^2+1=3(x^2-1);\,\)$ $\(x^2+1=3x^2-3;\,\)$

$\(2x^2=4;\,\)$ $\(x^2=2;\,\)$ $\(x_3=\sqrt{2};\,x_4=-\sqrt{2}.\)$

Ответ: $\(x_1=\sqrt{3};\,x_2=-\sqrt{3};\,x_3=\sqrt{2};\,x_4=-\sqrt{2}.\)$

Answer 2

To solve the equation $\frac{{x^2+1}}{{x^2-1}}+\frac{{6(x^2-1)}}{{x^2+1}}-5=0$ using a new variable, we can let $u = x^2 + 1$. This substitution simplifies the equation and makes it easier to solve for $u$ and then find the corresponding values of $x$. Let's proceed step by step:

Step 1: Substitute $u = x^2 + 1$:

After the substitution, the equation becomes: $\frac{u}{x^2-1} + \frac{6(x^2-1)}{u} - 5 = 0$.

Step 2: Simplify the expression:

To get rid of the denominators, we can multiply the entire equation by $u \cdot (x^2 - 1)$:

$u^2 + 6(x^2-1)^2 - 5u \cdot (x^2-1) = 0$.

Step 3: Expand and collect like terms:

Expand $(x^2-1)^2$ and rearrange the equation:

$u^2 + 6(x^4 - 2x^2 + 1) - 5u \cdot (x^2 - 1) = 0$.

Step 4: Simplify further:

Expand the $6(x^4 - 2x^2 + 1)$ term:

$u^2 + 6x^4 - 12x^2 + 6 - 5u \cdot (x^2 - 1) = 0$.

Step 5: Combine like terms:

Rearrange the terms to group the $u^2$ and $u$ terms together:

$u^2 - 5u \cdot (x^2 - 1) + 6x^4 - 12x^2 + 6 = 0$.

Step 6: Factor the quadratic in terms of $u$:

The equation is now in the form $au^2 + bu + c = 0$ where $a = 1$, $b = -5 \cdot (x^2 - 1)$, and $c = 6x^4 - 12x^2 + 6$.

Apply the quadratic formula to solve for $u$:

$u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Substitute the values of $a$, $b$, and $c$:

$u = \frac{5(x^2 - 1) \pm \sqrt{(5(x^2 - 1))^2 - 4 \cdot 1 \cdot (6x^4 - 12x^2 + 6)}}{2 \cdot 1}$.

Simplify the expression under the square root:

$u = \frac{5(x^2 - 1) \pm \sqrt{25(x^2 - 1)^2 - 24x^4 + 48x^2 - 24}}{2}$.

Step 7: Further simplify and solve for $u$:

$u = \frac{5(x^2 - 1) \pm \sqrt{25(x^2 - 1)^2 - 24(x^4 - 2x^2 + 1)}}{2}$