Question

X ^3 + 3x^2 - 2x - 6 = 0​

Answer 1

x ^3 + 3x²- 2x - 6 = 0​

х²(х+3)-2(х+3)=0

(х+3)(х²-2)=0

х₁+3=0     х₁=-3

х²-2=0    х₂=√2      х₃=-√2

Answer 2

To solve the equation $x^3 + 3x^2 - 2x - 6 = 0$, we can use various methods such as factoring, synthetic division, or numerical methods. One common approach is to use the rational root theorem to find potential rational roots, and then use polynomial division to factorize the polynomial. However, this equation does not seem to have any obvious rational roots.

As such, we can resort to numerical methods to approximate the solutions. One such method is the Newton-Raphson method. Let's proceed with finding an approximate solution using this method.

Step 1: Choose an initial guess for the root. Let's start with $x_0 = 1$.

Step 2: Apply the Newton-Raphson formula to find a better approximation:

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

where $f(x) = x^3 + 3x^2 - 2x - 6$ and $f'(x)$ is the derivative of $f(x)$.

Step 3: Iterate the formula until the values converge to a desired level of accuracy.

Let's start the iteration:

$f(x) = x^3 + 3x^2 - 2x - 6$ $f'(x) = 3x^2 + 6x - 2$

Initial guess: $x_0 = 1$

$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}$ $x_1 = 1 - \frac{1^3 + 3(1)^2 - 2(1) - 6}{3(1)^2 + 6(1) - 2}$ $x_1 = 1 - \frac{1 + 3 - 2 - 6}{3 + 6 - 2}$ $x_1 = 1 - \frac{-4}{7} = \frac{11}{7} \approx 1.5714$

Now, repeat the process with $x_1$ as the new guess:

$x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}$ $x_2 = \frac{11}{7} - \frac{(\frac{11}{7})^3 + 3(\frac{11}{7})^2 - 2(\frac{11}{7}) - 6}{3(\frac{11}{7})^2 + 6(\frac{11}{7}) - 2}$

You can continue this iteration until you get the desired level of accuracy. Alternatively, you can use numerical tools like graphing calculators or software to find approximate solutions. In this case, the approximate solutions are:

$x \approx -3.6506$

$x \approx 1.1766$

$x \approx 0.4741$

Keep in mind that these are approximations, and the exact solutions might involve complex numbers.