Solving X^3 + 2X^(3/2) = 0

Introduction

In mathematics, solving equations is the process of finding the values of the unknown variables that make the equation true. In this article, we will discuss how to solve the equation x^3 + 2x^(3/2) = 0, which involves both a cube and a square root.

Understanding the Equation

The equation x^3 + 2x^(3/2) = 0 can be rewritten as x^(3/2)(x + 2) = 0. This means that either x^(3/2) = 0 or x + 2 = 0. Solving for x^(3/2) = 0 gives us x = 0, while solving for x + 2 = 0 gives us x = -2.

Checking for Extraneous Solutions

When solving equations involving radicals, we must always check for extraneous solutions, which are solutions that do not actually satisfy the original equation. In this case, plugging x = 0 into the original equation gives us 0 + 2(0)^(3/2) = 0, which is true. Plugging x = -2 into the original equation gives us (-2)^3 + 2(-2)^(3/2) = -8 - 4√2, which is not equal to 0. Therefore, x = -2 is an extraneous solution.

The Solution

The only solution to the equation x^3 + 2x^(3/2) = 0 is x = 0.

Alternative Solution

Another way to solve the equation x^3 + 2x^(3/2) = 0 is to factor out x^(3/2), giving us x^(3/2)(x + 2) = 0. Then, we can divide both sides by x^(3/2), giving us x + 2 = 0 or x = -2. However, as we showed earlier, x = -2 is an extraneous solution, so the only solution is x = 0.

Graphical Solution

We can also solve the equation x^3 + 2x^(3/2) = 0 graphically. The graph of y = x^3 + 2x^(3/2) is shown below. We can see that the only x-intercept is at x = 0.

Graph of y = x^3 + 2x^(3/2)

Real-World Applications

The equation x^3 + 2x^(3/2) = 0 does not have any direct real-world applications. However, the process of solving equations is used in many fields, such as engineering, physics, and finance.

Conclusion

In this article, we discussed how to solve the equation x^3 + 2x^(3/2) = 0. We showed that the only solution is x = 0 and explained how to check for extraneous solutions. We also provided an alternative solution and a graphical solution. Finally, we discussed the importance of solving equations in various fields.