The Sum Of The Square Of A Number And One Less Than The Number Equals Nine Less Than The Square Of One More Than The Same Number. Find The Number.

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Introduction

Mathematics is a fascinating subject that deals with numbers, quantities, and shapes. It is a fundamental tool for problem-solving and critical thinking. In this article, we will delve into a mathematical problem that involves algebraic expressions and equations. The problem states that the sum of the square of a number and one less than the number equals nine less than the square of one more than the same number. Our goal is to find the number that satisfies this equation.

Understanding the Problem

Let's break down the problem statement and understand what it means. We are given an equation that involves a number, which we will call 'x'. The equation is:

x^2 + (x - 1) = (x + 1)^2 - 9

Our task is to find the value of 'x' that satisfies this equation.

Algebraic Manipulation

To solve this equation, we need to manipulate it algebraically. We can start by expanding the right-hand side of the equation using the formula (a + b)^2 = a^2 + 2ab + b^2.

(x + 1)^2 - 9 = x^2 + 2x + 1 - 9

Now, we can simplify the equation by combining like terms.

x^2 + (x - 1) = x^2 + 2x - 8

Simplifying the Equation

We can simplify the equation further by subtracting x^2 from both sides.

x - 1 = 2x - 8

Isolating the Variable

Next, we can isolate the variable 'x' by subtracting x from both sides.

-1 = x - 8

Solving for x

Finally, we can solve for 'x' by adding 8 to both sides.

x = 7

Conclusion

In this article, we have solved a mathematical problem that involves algebraic expressions and equations. We have manipulated the equation algebraically, simplified it, and isolated the variable 'x'. Our final answer is x = 7, which satisfies the given equation.

Real-World Applications

This problem may seem abstract, but it has real-world applications in various fields such as physics, engineering, and computer science. For example, in physics, the equation can be used to model the motion of an object under the influence of gravity. In engineering, the equation can be used to design and optimize systems such as bridges and buildings. In computer science, the equation can be used to develop algorithms for solving complex problems.

Tips and Tricks

Here are some tips and tricks for solving algebraic equations:

  • Always start by simplifying the equation and isolating the variable.
  • Use algebraic manipulation techniques such as expanding and combining like terms.
  • Check your work by plugging the solution back into the original equation.
  • Practice, practice, practice! The more you practice, the better you will become at solving algebraic equations.

Frequently Asked Questions

Q: What is the sum of the square of a number and one less than the number? A: The sum of the square of a number and one less than the number is x^2 + (x - 1).

Q: What is nine less than the square of one more than the same number? A: Nine less than the square of one more than the same number is (x + 1)^2 - 9.

Q: How do I solve the equation x^2 + (x - 1) = (x + 1)^2 - 9? A: To solve the equation, you can start by expanding the right-hand side of the equation using the formula (a + b)^2 = a^2 + 2ab + b^2. Then, you can simplify the equation by combining like terms and isolating the variable 'x'.

References

Q&A: The Sum of the Square of a Number and One Less Than the Number Equals Nine Less Than the Square of One More Than the Same Number

In this article, we will answer some of the most frequently asked questions about the problem "The sum of the square of a number and one less than the number equals nine less than the square of one more than the same number. Find the number."

Q: What is the sum of the square of a number and one less than the number?

A: The sum of the square of a number and one less than the number is x^2 + (x - 1).

Q: What is nine less than the square of one more than the same number?

A: Nine less than the square of one more than the same number is (x + 1)^2 - 9.

Q: How do I solve the equation x^2 + (x - 1) = (x + 1)^2 - 9?

A: To solve the equation, you can start by expanding the right-hand side of the equation using the formula (a + b)^2 = a^2 + 2ab + b^2. Then, you can simplify the equation by combining like terms and isolating the variable 'x'.

Q: What is the value of x that satisfies the equation?

A: The value of x that satisfies the equation is x = 7.

Q: Can you explain the steps to solve the equation in more detail?

A: Here are the steps to solve the equation:

  1. Expand the right-hand side of the equation using the formula (a + b)^2 = a^2 + 2ab + b^2.
  2. Simplify the equation by combining like terms.
  3. Isolate the variable 'x' by subtracting x^2 from both sides.
  4. Add 8 to both sides to solve for x.

Q: What are some real-world applications of this problem?

A: This problem has real-world applications in various fields such as physics, engineering, and computer science. For example, in physics, the equation can be used to model the motion of an object under the influence of gravity. In engineering, the equation can be used to design and optimize systems such as bridges and buildings. In computer science, the equation can be used to develop algorithms for solving complex problems.

Q: How can I practice solving algebraic equations like this one?

A: You can practice solving algebraic equations like this one by:

  • Working on algebraic expression and equation problems in your math textbook or online resources.
  • Using online tools and calculators to help you solve equations.
  • Practicing with real-world problems and applications.
  • Joining a study group or working with a tutor to get help and feedback.

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

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

  • Not simplifying the equation enough before solving for the variable.
  • Not isolating the variable correctly.
  • Not checking your work by plugging the solution back into the original equation.
  • Not practicing enough to build your skills and confidence.

Q: How can I improve my algebra skills?

A: You can improve your algebra skills by:

  • Practicing regularly and consistently.
  • Working on a variety of problems and applications.
  • Getting help and feedback from a tutor or study group.
  • Using online resources and tools to supplement your learning.
  • Reviewing and practicing concepts and formulas regularly.

Q: What are some additional resources for learning algebra?

A: Some additional resources for learning algebra include:

  • Online textbooks and resources such as Khan Academy, Mathway, and Wolfram Alpha.
  • Algebra textbooks and workbooks.
  • Online courses and tutorials.
  • Study groups and tutoring services.
  • Algebra apps and software.

Q: Can you provide more examples of algebraic equations like this one?

A: Here are a few more examples of algebraic equations:

  • x^2 + 2x = 3x - 5
  • x^2 - 4x = 2x - 3
  • x^2 + 3x = 2x + 5

These equations can be solved using the same steps as the original equation, and can be used to practice and build your skills in algebra.