Question

3-1/x²-1+10/1+x²=6/9

Answer 1

$3-\frac{1}{x^{2}-1}+\frac{10}{1+x^{2}}=\frac{6}{9}\\\\3-\frac{1}{x^{2}-1}+\frac{10}{x^{2}+1}-\frac{2}{3}=0\\\\\frac{9x^{4}-9-3x^{2}-3+30x^{2} -30-2x^{4}+2}{3(x^{2}-1)(x^{2}+1)}=0\\\\\frac{7x^{4}+27x^{2}-40}{3(x^{2}-1)(x^{2}+1)}=0 \\\\7x^{4} +27x^{2}-40=0;x\neq-1;x\neq1$

Сделаем замену : x² = m , m  ≥ 0

7m² + 27m -40 = 0

D = 27² - 4 * 7 * (-40) = 729 + 1120 = 1849 = 43²

$m_{1}=\frac{-27+43}{14}=\frac{16}{14}=\frac{8}{7}\\\\m_{2}=\frac{-27-43}{14}=-\frac{70}{14}=-5$

Answer 2

To solve the equation $\frac{3}{1} - \frac{x^2}{1} + \frac{10}{1+x^2} = \frac{6}{9}$, we need to simplify it and then find the value of $x$ that satisfies the equation.

Step 1: Combine the fractions on the left-hand side: The common denominator for the fractions is $1+x^2$. So, we can rewrite the equation as follows:

$\frac{3(1+x^2) - x^2(1+x^2) + 10}{1+x^2} = \frac{6}{9}$

Step 2: Simplify the left-hand side: Expand the numerator and combine like terms:

$\frac{3 + 3x^2 - x^2 - x^4 + 10}{1+x^2} = \frac{6}{9}$

$\frac{-x^4 + 3x^2 + 13}{1+x^2} = \frac{6}{9}$

Step 3: Simplify the right-hand side: Reduce the fraction on the right-hand side:

$\frac{-x^4 + 3x^2 + 13}{1+x^2} = \frac{2}{3}$

Step 4: Multiply both sides by $1+x^2$ to eliminate the denominator on the left-hand side:

$(-x^4 + 3x^2 + 13)(1+x^2) = \frac{2}{3}(1+x^2)$

Step 5: Expand and rearrange the equation:

$-x^6 + 3x^4 + 13x^2 + x^2 - \frac{2}{3}x^4 - \frac{2}{3}x^2 = \frac{2}{3}$

Step 6: Combine like terms:

$-x^6 + \left(3 - \frac{2}{3}\right)x^4 + \left(13 - \frac{2}{3}\right)x^2 = \frac{2}{3}$

$-x^6 + \frac{7}{3}x^4 + \frac{37}{3}x^2 = \frac{2}{3}$

Step 7: Bring all terms to one side of the equation to set it to zero:

$-x^6 + \frac{7}{3}x^4 + \frac{37}{3}x^2 - \frac{2}{3} = 0$

Step 8: The equation is now a sixth-degree polynomial, which may not have an easy analytical solution. You can use numerical methods or a graphing calculator to approximate the solutions of the equation. Alternatively, if this equation is part of a specific problem or context, you might want to check if there are particular restrictions on the values of $x$ or if there are any specific solutions required.

Without further context or constraints, I cannot determine the exact value(s) of $x$ that satisfy the equation.