Solve The Equation:${ 2h^3 - 11 = 43 }$

Mar 11, 2025 by ADMIN 41 views

Solving the Cubic Equation: A Step-by-Step Guide

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Introduction

In this article, we will delve into the world of cubic equations and explore a step-by-step solution to the given equation: $2h^3 - 11 = 43$ . Cubic equations are a fundamental concept in algebra, and understanding how to solve them is crucial for various applications in mathematics, physics, and engineering.

Understanding Cubic Equations

A cubic equation is a polynomial equation of degree three, which means the highest power of the variable (in this case, $h$ ) is three. The general form of a cubic equation is:

ax^3 + bx^2 + cx + d = 0

where $a$ , $b$ , $c$ , and $d$ are constants, and $x$ is the variable. In our given equation, $a = 2$ , $b = 0$ , $c = 0$ , and $d = -11$ .

Rearranging the Equation

To solve the equation, we first need to isolate the variable $h$ . We can do this by rearranging the equation to get:

2h^3 = 43 + 11

2h^3 = 54

Dividing by 2

Next, we divide both sides of the equation by 2 to get:

h^3 = 27

Taking the Cube Root

Now, we take the cube root of both sides of the equation to get:

h = \sqrt[3]{27}

Simplifying the Cube Root

The cube root of 27 can be simplified as:

h = \sqrt[3]{3^3}

h = 3

Verifying the Solution

To verify our solution, we can plug $h = 3$ back into the original equation:

2(3)^3 - 11 = 2(27) - 11

54 - 11 = 43

43 = 43

Conclusion

In this article, we have solved the cubic equation $2h^3 - 11 = 43$ using a step-by-step approach. We rearranged the equation, divided by 2, took the cube root, and simplified the cube root to find the solution $h = 3$ . We also verified our solution by plugging it back into the original equation.

Applications of Cubic Equations

Cubic equations have numerous applications in various fields, including:

Physics: Cubic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
Engineering: Cubic equations are used to design and optimize systems, such as bridges, buildings, and mechanical systems.
Computer Science: Cubic equations are used in computer graphics, game development, and scientific simulations.

Tips and Tricks for Solving Cubic Equations

Here are some tips and tricks for solving cubic equations:

Use the rational root theorem: This theorem 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.
Use synthetic division: This method allows you to divide a polynomial by a linear factor, which can help you find the roots of the equation.
Use numerical methods: If the equation is difficult to solve analytically, you can use numerical methods, such as the Newton-Raphson method, to approximate the roots.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving cubic equations:

Not checking for extraneous solutions: Make sure to check your solutions by plugging them back into the original equation.
Not using the correct method: Choose the correct method for solving the equation, such as factoring, synthetic division, or numerical methods.
Not verifying the solution: Verify your solution by plugging it back into the original equation.

Conclusion

Solving cubic equations requires a step-by-step approach, and understanding the different methods and techniques can help you solve these equations with confidence. By following the tips and tricks outlined in this article, you can avoid common mistakes and find the solutions to cubic equations.

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Q: What is a cubic equation?

A: A cubic equation is a polynomial equation of degree three, which means the highest power of the variable (in this case, $h$ ) is three. The general form of a cubic equation is:

ax^3 + bx^2 + cx + d = 0

Q: How do I solve a cubic equation?

A: To solve a cubic equation, you can use various methods, such as:

Factoring: If the equation can be factored, you can set each factor equal to zero and solve for the variable.
Synthetic division: This method allows you to divide a polynomial by a linear factor, which can help you find the roots of the equation.
Numerical methods: If the equation is difficult to solve analytically, you can use numerical methods, such as the Newton-Raphson method, to approximate the roots.

Q: What is the rational root theorem?

A: The rational root theorem 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.

Q: How do I use synthetic division?

A: To use synthetic division, you need to:

Divide the polynomial by a linear factor: This can be done using the synthetic division algorithm.
Find the remainder: The remainder is the value that is left after dividing the polynomial by the linear factor.
Use the remainder to find the roots: The remainder can be used to find the roots of the equation.

Q: What is the Newton-Raphson method?

A: The Newton-Raphson method is a numerical method used to approximate the roots of an equation. It works by:

Starting with an initial guess: You need to start with an initial guess for the root.
Iterating: You need to iterate the formula until you get the desired level of accuracy.

Q: How do I verify my solution?

A: To verify your solution, you need to:

Plug the solution back into the original equation: This will help you check if the solution is correct.
Check for extraneous solutions: Make sure to check for extraneous solutions, which are solutions that are not valid.

Q: What are some common mistakes to avoid?

A: Some common mistakes to avoid when solving cubic equations include:

Not checking for extraneous solutions: Make sure to check your solutions by plugging them back into the original equation.
Not using the correct method: Choose the correct method for solving the equation, such as factoring, synthetic division, or numerical methods.
Not verifying the solution: Verify your solution by plugging it back into the original equation.

Q: How do I apply cubic equations in real-life situations?

A: Cubic equations have numerous applications in various fields, including:

Physics: Cubic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
Engineering: Cubic equations are used to design and optimize systems, such as bridges, buildings, and mechanical systems.
Computer Science: Cubic equations are used in computer graphics, game development, and scientific simulations.

Q: What are some tips for solving cubic equations?

A: Some tips for solving cubic equations include:

Use the rational root theorem: This theorem 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.
Use synthetic division: This method allows you to divide a polynomial by a linear factor, which can help you find the roots of the equation.
Use numerical methods: If the equation is difficult to solve analytically, you can use numerical methods, such as the Newton-Raphson method, to approximate the roots.

Q: How do I know if I have found the correct solution?

A: To know if you have found the correct solution, you need to:

Verify your solution: Verify your solution by plugging it back into the original equation.
Check for extraneous solutions: Make sure to check for extraneous solutions, which are solutions that are not valid.

Q: What are some resources for learning more about cubic equations?

A: Some resources for learning more about cubic equations include:

Textbooks: There are many textbooks available that cover cubic equations in detail.
Online resources: There are many online resources available, such as Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
Mathematical software: There are many mathematical software packages available, such as Mathematica, Maple, and MATLAB, that can help you solve cubic equations.