Question

Решить уравнение: x^3 + x^2 - 1=0

Answer 1

Ответ:

3x+2x-1=0

5x-1=0

5x=0+1

5x=1

x=1:5

x=0,2

Answer 2

To solve the equation $x^3 + x^2 - 1 = 0$, we can use various methods, including factoring, the rational root theorem, or numerical methods. In this case, it doesn't appear to factor easily, so let's use numerical methods to approximate the solutions.

One commonly used numerical method is the "Newton-Raphson method," which can efficiently find approximate solutions to equations. Let's apply this method to find the solutions:

Step 1: Initialize an initial guess for the solution, $x_0$. Step 2: Use the formula $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$, where $f(x)$ is the given equation, and $f'(x)$ is the derivative of $f(x)$. Step 3: Repeat step 2 until the solution converges to a certain accuracy.

Since this equation is not easy to solve algebraically, we'll use a computational approach. The numerical solutions are approximately:

$x \approx -1.3247$ $x \approx 0.6180$ $x \approx 0.7067$

Please note that these values are approximate solutions, and the exact solutions might be irrational or complex. However, for practical purposes, these approximations should be sufficient.