X2+2x3-3x-6=0

To solve the equation \( x^2 + 2x^3 - 3x - 6 = 0 \), we can start by rearranging it into standard form:

\[ 2x^3 + x^2 - 3x - 6 = 0 \]

Next, we can try to find rational roots using the Rational Root Theorem. Potential rational roots would be the factors of the constant term (\(-6\)) divided by the factors of the leading coefficient (\(2\)). The possible rational roots are:

\[ \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2} \]

Let's test these possible roots by substituting them into the equation:

1. \( x = 1 \): \[ 2(1)^3 + (1)^2 - 3(1) - 6 = 2 + 1 - 3 - 6 = -6 \quad (\text{not a root}) \]

2. \( x = -1 \): \[ 2(-1)^3 + (-1)^2 - 3(-1) - 6 = -2 + 1 + 3 - 6 = -4 \quad (\text{not a root}) \]

3. \( x = 2 \): \[ 2(2)^3 + (2)^2 - 3(2) - 6 = 2(8) + 4 - 6 - 6 = 16 + 4 - 6 - 6 = 8 \quad (\text{not a root}) \]

4. \( x = -2 \): \[ 2(-2)^3 + (-2)^2 - 3(-2) - 6 = 2(-8) + 4 + 6 - 6 = -16 + 4 + 6 - 6 = -12 \quad (\text{not a root}) \]

5. \( x = 3 \): \[ 2(3)^3 + (3)^2 - 3(3) - 6 = 2(27) + 9 - 9 - 6 = 54 + 9 - 9 - 6 = 48 \quad (\text{not a root}) \]

6. \( x = -3 \): \[ 2(-3)^3 + (-3)^2 - 3(-3) - 6 = 2(-27) + 9 + 9 - 6 = -54 + 9 + 9 - 6 = -42 \quad (\text{not a root}) \]

7. \( x = 6 \): \[ 2(6)^3 + (6)^2 - 3(6) - 6 = 2(216) + 36 - 18 - 6 = 432 + 36 - 18 - 6 = 444 \quad (\text{not a root}) \]

8. \( x = -6 \): \[ 2(-6)^3 + (-6)^2 - 3(-6) - 6 = 2(-216) + 36 + 18 - 6 = -432 + 36 + 18 - 6 = -384 \quad (\text{not a root}) \]

9. \( x = \frac{1}{2} \): \[ 2\left(\frac{1}{2}\right)^3 + \left(\frac{1}{2}\right)^2 - 3\left(\frac{1}{2}\right) - 6 = 2\left(\frac{1}{8}\right) + \frac{1}{4} - \frac{3}{2} - 6 = \frac{1}{4} + \frac{1}{4} - \frac{3}{2} - 6 = \frac{1}{2} - \frac{3}{2} - 6 = -6 \quad (\text{not a root}) \]

10. \( x = -\frac{1}{2} \): \[ 2\left(-\frac{1}{2}\right)^3 + \left(-\frac{1}{2}\right)^2 - 3\left(-\frac{1}{2}\right) - 6 = 2\left(-\frac{1}{8}\right) + \frac{1}{4} + \frac{3}{2} - 6 = -\frac{1}{4} + \frac{1}{4} + \frac{3}{2} - 6 = -6 \quad (\text{not a root}) \]

11. \( x = \frac{3}{2} \): \[ 2\left(\frac{3}{2}\right)^3 + \left(\frac{3}{2}\right)^2 - 3\left(\frac{3}{2}\right) - 6 = 2\left(\frac{27}{8}\right) + \frac{9}{4} - \frac{9}{2} - 6 = \frac{27}{4} + \frac{9}{4} - \frac{18}{4} - \frac{24}{4} = -6 \quad (\text{not a root}) \]

12. \( x = -\frac{3}{2} \): \[ 2\left(-\frac{3}{2}\right)^3 + \left(-\frac{3}{2}\right)^2 - 3\left(-\frac{3}{2}\right) - 6 = 2\left(-\frac{27}{8}\right) + \frac{9}{4} + \frac{9}{2} - 6 = -\frac{27}{4} + \frac{9}{4} + \frac{18}{4} - \frac{24}{4} = -6 \quad (\text{not a root}) \]

After testing rational roots, none of the possible values worked.

Since we have not found any rational roots, we may want to resort to numerical methods or graphing methods to find approximate solutions or employ synthetic division or polynomial long division to factor it further.

If you're looking for a numerical solution, you can use numerical methods, such as Newton's method or using graphing calculators to approximate the real roots.

Would you like guidance on graphing or numerical methods for further solving?