Solve The Equation: $(x+2)^3 - 43 = 300$

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Introduction

In this article, we will delve into the world of algebra and focus on solving a cubic equation. The given equation is $(x+2)^3 - 43 = 300$, and our goal is to find the value of xx that satisfies this equation. We will break down the solution into manageable steps, making it easier to understand and follow along.

Understanding the Equation

The given equation is a cubic equation, which means it has a variable raised to the power of 3. In this case, the variable is xx, and it is raised to the power of 3 inside the parentheses. The equation can be rewritten as $(x+2)^3 = 300 + 43$, which simplifies to $(x+2)^3 = 343$.

Isolating the Variable

To solve for xx, we need to isolate the variable on one side of the equation. We can start by taking the cube root of both sides of the equation. This will give us x+2=3433x+2 = \sqrt[3]{343}.

Simplifying the Cube Root

The cube root of 343 is 7, since 73=3437^3 = 343. Therefore, we can simplify the equation to x+2=7x+2 = 7.

Solving for x

To solve for xx, we need to isolate the variable on one side of the equation. We can do this by subtracting 2 from both sides of the equation. This gives us x=7−2x = 7 - 2, which simplifies to x=5x = 5.

Verifying the Solution

To verify that our solution is correct, we can plug it back into the original equation. Substituting x=5x = 5 into the equation, we get $(5+2)^3 - 43 = 300$. Simplifying this expression, we get $(7)^3 - 43 = 300$, which further simplifies to 343−43=300343 - 43 = 300. This is indeed true, so we can confirm that our solution is correct.

Conclusion

In this article, we solved the cubic equation $(x+2)^3 - 43 = 300$ by breaking it down into manageable steps. We isolated the variable, simplified the cube root, and solved for xx. We then verified our solution by plugging it back into the original equation. The final answer is x=5x = 5.

Tips and Tricks

  • When solving cubic equations, it's often helpful to start by isolating the variable on one side of the equation.
  • Simplifying the cube root can make it easier to solve for the variable.
  • Verifying the solution by plugging it back into the original equation can help ensure that the solution is correct.

Real-World Applications

Cubic equations have many real-world applications, including:

  • Physics: Cubic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Cubic equations are used to design and optimize systems, such as bridges and buildings.
  • Computer Science: Cubic equations are used in algorithms and data structures to solve problems efficiently.

Common Mistakes

  • Not isolating the variable: Failing to isolate the variable on one side of the equation can make it difficult to solve for the variable.
  • Not simplifying the cube root: Failing to simplify the cube root can make it difficult to solve for the variable.
  • Not verifying the solution: Failing to verify the solution by plugging it back into the original equation can lead to incorrect solutions.

Conclusion

Solving cubic equations requires a combination of algebraic skills and problem-solving strategies. By breaking down the equation into manageable steps, simplifying the cube root, and verifying the solution, we can find the value of xx that satisfies the equation. The final answer is x=5x = 5.

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Q: What is a cubic equation?

A: A cubic equation is a polynomial equation of degree three, which means it has a variable raised to the power of 3. The general form of a cubic equation is ax3+bx2+cx+d=0ax^3 + bx^2 + cx + d = 0, where aa, bb, cc, and dd are constants.

Q: How do I solve a cubic equation?

A: To solve a cubic equation, you can start by isolating the variable on one side of the equation. Then, simplify the cube root and solve for the variable. Finally, verify the solution by plugging it back into the original equation.

Q: What is the difference between a cubic equation and a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means it has a variable raised to the power of 2. The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants. Cubic equations are more complex and require different solution methods.

Q: Can I use a calculator to solve a cubic equation?

A: Yes, you can use a calculator to solve a cubic equation. However, it's often helpful to understand the underlying algebraic methods and to verify the solution by plugging it back into the original equation.

Q: What are some common mistakes to avoid when solving cubic equations?

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

  • Not isolating the variable on one side of the equation
  • Not simplifying the cube root
  • Not verifying the solution by plugging it back into the original equation

Q: Can I use a graphing calculator to solve a cubic equation?

A: Yes, you can use a graphing calculator to solve a cubic equation. Graphing calculators can help you visualize the solution and find the roots of the equation.

Q: What are some real-world applications of cubic equations?

A: Cubic equations have many real-world applications, including:

  • Physics: Cubic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Cubic equations are used to design and optimize systems, such as bridges and buildings.
  • Computer Science: Cubic equations are used in algorithms and data structures to solve problems efficiently.

Q: Can I use a computer program to solve a cubic equation?

A: Yes, you can use a computer program to solve a cubic equation. Many computer algebra systems, such as Mathematica and Maple, can solve cubic equations and provide a detailed solution.

Q: What is the difference between a cubic equation and a polynomial equation of degree three?

A: A cubic equation is a polynomial equation of degree three, but it is not necessarily a polynomial equation of degree three with integer coefficients. A polynomial equation of degree three is a more general term that includes cubic equations, but also other types of polynomial equations of degree three.

Q: Can I use a substitution method to solve a cubic equation?

A: Yes, you can use a substitution method to solve a cubic equation. This involves substituting a new variable into the equation to simplify it and make it easier to solve.

Q: What are some tips for solving cubic equations?

A: Some tips for solving cubic equations include:

  • Start by isolating the variable on one side of the equation
  • Simplify the cube root
  • Verify the solution by plugging it back into the original equation
  • Use a calculator or computer program to check your solution
  • Practice solving cubic equations to build your skills and confidence.