To solve the cubic equation $x^3 + 2x + 5 = 0$, follow these steps:

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Step 1: Confirm the General Form

The equation is already in the form $x^3 + bx + c = 0$.

Here:

• $a = 1$,
• $b = 2$,
• $c = 5$.

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Step 2: Analyze the Equation

This is a cubic equation, so it may have one real root and two complex conjugate roots. Finding the roots of a cubic equation often requires numerical or approximation methods since closed-form solutions are generally cumbersome.

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Step 3: Apply Numerical Methods

Let's approximate the real root using the Newton-Raphson method or a computational approach. Alternatively, a general approach involves the Cardano's formula, which can be complex to compute manually. Let’s compute the approximate solutions.

from sympy import symbols, Eq, solve

Define the variable and the equation

x = symbols('x') equation = Eq(x**3 + 2*x + 5, 0)

Solve the cubic equation numerically

solutions = solve(equation, x) solutions[2/((-1/2 - sqrt(3)I/2)(135/2 + 3sqrt(2121)/2)**(1/3)) - (-1/2 - sqrt(3)I/2)(135/2 + 3sqrt(2121)/2)(1/3)/3, -(-1/2 + sqrt(3)I/2)(135/2 + 3*sqrt(2121)/2)(1/3)/3 + 2/((-1/2 + sqrt(3)I/2)(135/2 + 3sqrt(2121)/2)**(1/3)), -(135/2 + 3sqrt(2121)/2)(1/3)/3 + 2/(135/2 + 3*sqrt(2121)/2)(1/3)][2/((-1/2 - sqrt(3)I/2)(135/2 + 3sqrt(2121)/2)**(1/3)) - (-1/2 - sqrt(3)I/2)(135/2 + 3sqrt(2121)/2)(1/3)/3, -(-1/2 + sqrt(3)I/2)(135/2 + 3*sqrt(2121)/2)(1/3)/3 + 2/((-1/2 + sqrt(3)I/2)(135/2 + 3sqrt(2121)/2)**(1/3)), -(135/2 + 3sqrt(2121)/2)(1/3)/3 + 2/(135/2 + 3*sqrt(2121)/2)(1/3)]successThe roots of the equation $x^3 + 2x + 5 = 0$ are complex and include one real root and two complex conjugate roots. The exact symbolic solutions are:

1. $x_1 = \frac{2}{\left(-\frac{1}{2} - \frac{\sqrt{3}i}{2}\right)\left(\frac{135}{2} + \frac{3\sqrt{2121}}{2}\right)^{1/3}} - \frac{\left(-\frac{1}{2} - \frac{\sqrt{3}i}{2}\right)\left(\frac{135}{2} + \frac{3\sqrt{2121}}{2}\right)^{1/3}}{3}$

2. $x_2 = \frac{2}{\left(-\frac{1}{2} + \frac{\sqrt{3}i}{2}\right)\left(\frac{135}{2} + \frac{3\sqrt{2121}}{2}\right)^{1/3}} - \frac{\left(-\frac{1}{2} + \frac{\sqrt{3}i}{2}\right)\left(\frac{135}{2} + \frac{3\sqrt{2121}}{2}\right)^{1/3}}{3}$

3. $x_3 = -\frac{\left(\frac{135}{2} + \frac{3\sqrt{2121}}{2}\right)^{1/3}}{3} + \frac{2}{\left(\frac{135}{2} + \frac{3\sqrt{2121}}{2}\right)^{1/3}}$

These solutions are exact but involve complex expressions. Numerical approximations can make them more interpretable.

Would you like a numerical approximation for these roots or a detailed explanation of their symbolic derivation?

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Related Questions:

1. How does Cardano's formula work for cubic equations?
2. What numerical methods can approximate roots efficiently?
3. How can we determine the nature of roots (real or complex)?
4. What is the significance of the discriminant in cubic equations?
5. Can this equation model any real-world scenarios?

Tip:

When solving cubic equations, approximate solutions are often sufficient, especially in real-world applications.