Expressing the Solution in Trigonometric Form

Oct 13, 2025 by miodeya.com 27 views

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Introduction

Cubic equations are a fundamental concept in algebra, and solving them can be a challenging task. In this article, we will explore a specific cubic equation and provide a step-by-step guide on how to solve it. We will also discuss the different solutions to the equation and provide explanations for each.

The Cubic Equation

The given cubic equation is:

$2x^3 - 64 = x^3 + 61$

Our goal is to find the solution to this equation, which means finding the value of $x$ that satisfies the equation.

Rearranging the Equation

To solve the equation, we first need to rearrange it to get all the terms on one side. We can do this by subtracting $x^3$ from both sides and adding $64$ to both sides:

$2x^3 - x^3 = 64 + 61$

Simplifying the left-hand side, we get:

$x^3 = 125$

Finding the Solution

Now that we have the equation $x^3 = 125$ , we can find the solution by taking the cube root of both sides:

$x = \sqrt[3]{125}$

Simplifying the Cube Root

The cube root of $125$ can be simplified as follows:

$\sqrt[3]{125} = \sqrt[3]{5^3} = 5$

However, this is not the only solution to the equation. We can also express the cube root of $125$ in terms of complex numbers:

$\sqrt[3]{125} = 5 \left( \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} \right)$

Expressing the Solution in Trigonometric Form

Using De Moivre's theorem, we can express the solution in trigonometric form as:

$x = 5 \left( \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} \right)$

Converting to Complex Number Form

We can convert the trigonometric form to complex number form using Euler's formula:

$x = 5 \left( \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} \right) = 5 \left( \frac{-1 + i\sqrt{3}}{2} \right)$

Simplifying the Complex Number

Simplifying the complex number, we get:

$x = \frac{-5 + 5i\sqrt{3}}{2}$

Conclusion

In this article, we have solved a cubic equation and found the solution in complex number form. We have also discussed the different solutions to the equation and provided explanations for each. The solution to the equation is $\frac{-5 + 5i\sqrt{3}}{2}$ .

Answer

The correct answer is:

B) $\frac{-5 + 5i\sqrt{3}}{2}$

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Introduction

In our previous article, we solved a cubic equation and found the solution in complex number form. In this article, we will answer some frequently asked questions related to cubic equations and provide additional information to help you better understand the concept.

Q: What is a cubic equation?

A: A cubic equation is a polynomial equation of degree three, which means it has the highest power of the variable (in this case, x) equal to three. Cubic equations are often written in the form ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants.

Q: How do I solve a cubic equation?

A: Solving a cubic equation can be a challenging task, but there are several methods you can use. One common method is to use the rational root theorem, which states that if a rational number p/q is a root of the equation, then p must be a factor of the constant term and q must be a factor of the leading coefficient. You can also use synthetic division or numerical methods to solve cubic equations.

Q: What is the difference between a real and complex solution?

A: A real solution is a solution that can be expressed as a real number, whereas a complex solution is a solution that involves imaginary numbers. In the case of the cubic equation we solved earlier, the solution is a complex number, which means it involves imaginary numbers.

Q: How do I convert a complex number to trigonometric form?

A: To convert a complex number to trigonometric form, you can use Euler's formula, which states that e^(ix) = cos(x) + i sin(x). You can also use the polar form of a complex number, which is given by r(cos(x) + i sin(x)), where r is the magnitude of the complex number and x is the argument.

Q: What is the significance of the cube root of unity?

A: The cube root of unity is a complex number that satisfies the equation x^3 = 1. It is often denoted by ω and is used in many mathematical and scientific applications, including algebra, geometry, and physics.

Q: Can you provide more examples of cubic equations?

A: Yes, here are a few more examples of cubic equations:

x^3 - 6x^2 + 11x - 6 = 0
x^3 + 2x^2 - 5x - 2 = 0
x^3 - 3x^2 - 4x + 12 = 0

You can try solving these equations using the methods we discussed earlier.

Q: Are there any online resources that can help me learn more about cubic equations?

A: Yes, there are many online resources that can help you learn more about cubic equations, including:

Khan Academy: Cubic Equations
Mathway: Cubic Equations
Wolfram Alpha: Cubic Equations
MIT OpenCourseWare: Cubic Equations

These resources provide a wealth of information and examples to help you learn more about cubic equations.

Conclusion

In this article, we have answered some frequently asked questions related to cubic equations and provided additional information to help you better understand the concept. We hope this article has been helpful in your learning journey. If you have any more questions or need further clarification, please don't hesitate to ask.