Which Equation Is Equivalent To The Given Equation?Given Equation: X 2 + 27 = − 12 X X^2 + 27 = -12x X 2 + 27 = − 12 X A. X 2 + 12 X − 27 = 0 X^2 + 12x - 27 = 0 X 2 + 12 X − 27 = 0 B. X 2 − 12 X − 27 = 0 X^2 - 12x - 27 = 0 X 2 − 12 X − 27 = 0 C. X 2 + 12 X + 27 = 0 X^2 + 12x + 27 = 0 X 2 + 12 X + 27 = 0 D. X 2 − 12 X + 27 = 0 X^2 - 12x + 27 = 0 X 2 − 12 X + 27 = 0

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Understanding the Given Equation

The given equation is x2+27=−12xx^2 + 27 = -12x. To find an equivalent equation, we need to manipulate the given equation to match one of the options provided. The first step is to isolate the variable xx on one side of the equation.

Step 1: Rearrange the Equation

We can start by moving the −12x-12x term to the left-hand side of the equation and the constant term to the right-hand side. This gives us:

x2+12x+27=0x^2 + 12x + 27 = 0

Step 2: Compare with the Options

Now, let's compare the rearranged equation with the options provided:

  • A. x2+12x−27=0x^2 + 12x - 27 = 0
  • B. x2−12x−27=0x^2 - 12x - 27 = 0
  • C. x2+12x+27=0x^2 + 12x + 27 = 0
  • D. x2−12x+27=0x^2 - 12x + 27 = 0

Which Equation is Equivalent?

Comparing the rearranged equation with the options, we can see that option C matches the equation we obtained in Step 1.

Why is Option C Equivalent?

Option C is equivalent to the given equation because we moved the −12x-12x term to the left-hand side and the constant term to the right-hand side. This is a valid algebraic manipulation that preserves the equality of the equation.

Conclusion

In conclusion, the equivalent equation to the given equation x2+27=−12xx^2 + 27 = -12x is option C: x2+12x+27=0x^2 + 12x + 27 = 0.

Understanding the Algebraic Manipulation

The algebraic manipulation involved in rearranging the equation is a fundamental concept in mathematics. It involves moving terms from one side of the equation to the other while preserving the equality of the equation.

Why is Algebraic Manipulation Important?

Algebraic manipulation is essential in mathematics because it allows us to simplify equations, solve for variables, and make predictions about the behavior of mathematical models. By understanding how to manipulate equations, we can gain insights into the underlying structure of mathematical relationships.

Real-World Applications of Algebraic Manipulation

Algebraic manipulation has numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, in physics, algebraic manipulation is used to describe the motion of objects under the influence of forces. In engineering, algebraic manipulation is used to design and optimize systems. In economics, algebraic manipulation is used to model and analyze economic systems.

Conclusion

In conclusion, the equivalent equation to the given equation x2+27=−12xx^2 + 27 = -12x is option C: x2+12x+27=0x^2 + 12x + 27 = 0. Algebraic manipulation is a fundamental concept in mathematics that has numerous real-world applications.

Step-by-Step Solution

  1. Rearrange the Equation: Move the −12x-12x term to the left-hand side and the constant term to the right-hand side.
  2. Compare with the Options: Compare the rearranged equation with the options provided.
  3. Identify the Equivalent Equation: Identify the option that matches the rearranged equation.

Key Concepts

  • Algebraic manipulation
  • Rearranging equations
  • Equivalent equations
  • Real-world applications of algebraic manipulation

Tips and Tricks

  • Pay attention to the signs of the terms when rearranging the equation.
  • Use algebraic manipulation to simplify equations and solve for variables.
  • Practice rearranging equations to develop your problem-solving skills.

Common Mistakes

  • Failing to move the constant term to the right-hand side.
  • Failing to move the variable term to the left-hand side.
  • Not paying attention to the signs of the terms.

Real-World Examples

  • Physics: Algebraic manipulation is used to describe the motion of objects under the influence of forces.
  • Engineering: Algebraic manipulation is used to design and optimize systems.
  • Economics: Algebraic manipulation is used to model and analyze economic systems.

Conclusion

Frequently Asked Questions

Q: What is algebraic manipulation?

A: Algebraic manipulation is the process of rearranging and simplifying equations to make them easier to solve or understand. It involves moving terms from one side of the equation to the other while preserving the equality of the equation.

Q: Why is algebraic manipulation important?

A: Algebraic manipulation is essential in mathematics because it allows us to simplify equations, solve for variables, and make predictions about the behavior of mathematical models. By understanding how to manipulate equations, we can gain insights into the underlying structure of mathematical relationships.

Q: How do I rearrange an equation?

A: To rearrange an equation, you need to move the terms from one side of the equation to the other while preserving the equality of the equation. This involves moving the variable terms to one side and the constant terms to the other side.

Q: What is an equivalent equation?

A: An equivalent equation is an equation that has the same solution as the original equation. It is obtained by rearranging the original equation through algebraic manipulation.

Q: How do I identify an equivalent equation?

A: To identify an equivalent equation, you need to compare the rearranged equation with the options provided. Look for the option that matches the rearranged equation.

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

A: Some common mistakes to avoid when rearranging equations include:

  • Failing to move the constant term to the right-hand side.
  • Failing to move the variable term to the left-hand side.
  • Not paying attention to the signs of the terms.

Q: What are some real-world applications of algebraic manipulation?

A: Algebraic manipulation has numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, in physics, algebraic manipulation is used to describe the motion of objects under the influence of forces. In engineering, algebraic manipulation is used to design and optimize systems. In economics, algebraic manipulation is used to model and analyze economic systems.

Q: How can I practice rearranging equations?

A: You can practice rearranging equations by working through examples and exercises. Start with simple equations and gradually move on to more complex ones. Pay attention to the signs of the terms and make sure to move the variable terms to one side and the constant terms to the other side.

Q: What are some tips and tricks for rearranging equations?

A: Some tips and tricks for rearranging equations include:

  • Pay attention to the signs of the terms.
  • Use algebraic manipulation to simplify equations and solve for variables.
  • Practice rearranging equations to develop your problem-solving skills.

Q: What are some common misconceptions about algebraic manipulation?

A: Some common misconceptions about algebraic manipulation include:

  • Thinking that algebraic manipulation is only for solving equations.
  • Thinking that algebraic manipulation is only for simple equations.
  • Not realizing that algebraic manipulation is a fundamental concept in mathematics.

Q: How can I apply algebraic manipulation in real-world situations?

A: You can apply algebraic manipulation in real-world situations by using it to describe the motion of objects under the influence of forces, design and optimize systems, model and analyze economic systems, and solve problems in physics, engineering, economics, and computer science.

Q: What are some advanced topics in algebraic manipulation?

A: Some advanced topics in algebraic manipulation include:

  • Solving systems of equations.
  • Solving quadratic equations.
  • Solving polynomial equations.
  • Using algebraic manipulation to solve problems in calculus and differential equations.

Q: How can I learn more about algebraic manipulation?

A: You can learn more about algebraic manipulation by:

  • Reading textbooks and online resources.
  • Working through examples and exercises.
  • Practicing rearranging equations.
  • Seeking help from teachers or tutors.

Conclusion

In conclusion, algebraic manipulation is a fundamental concept in mathematics that has numerous real-world applications. By understanding how to manipulate equations, we can gain insights into the underlying structure of mathematical relationships.