Subtracting 2x from Both Sides

Oct 13, 2025 by recoveryshake.com 104 views

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 states that the sum of the square of x and one less than x equals nine less than the square of one more than x. Mathematically, this can be represented as:

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

Simplifying the Equation

To solve this equation, we need to simplify it and isolate the variable x. Let's start by expanding the right-hand side of the equation:

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

Now, substitute this expression into the original equation:

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

Combining Like Terms

Next, let's combine like terms on both sides of the equation:

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

Subtracting x^2 from Both Sides

Now, let's subtract x^2 from both sides of the equation to eliminate the x^2 term:

x - 1 = 2x - 8

Subtracting 2x from Both Sides

Next, let's subtract 2x from both sides of the equation to isolate the x term:

-x - 1 = -8

Adding 1 to Both Sides

Now, let's add 1 to both sides of the equation to get rid of the negative term:

-x = -7

Multiplying Both Sides by -1

Finally, let's multiply both sides of the equation by -1 to solve for x:

x = 7

Conclusion

In this article, we have solved a mathematical problem that involves algebraic expressions and equations. We have simplified the equation, combined like terms, and isolated the variable x to find the number that satisfies the equation. The solution to the problem is x = 7.

Real-World Applications

This problem may seem abstract and unrelated to real-world applications, but it has several practical uses. For example, in physics, the equation of motion can be represented as a quadratic equation, which can be solved using algebraic methods. In engineering, the design of structures such as bridges and buildings requires the use of mathematical models, which can be represented as algebraic equations.

Future Research Directions

This problem has several extensions and variations that can be explored in future research. For example, we can consider the case where the equation is not quadratic, but rather a higher-degree polynomial. We can also consider the case where the equation is not a polynomial, but rather a transcendental equation. Additionally, we can explore the use of numerical methods to solve the equation, such as the Newton-Raphson method.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by Michael Spivak
[3] "Linear Algebra and Its Applications" by Gilbert Strang

Glossary

Algebraic expression: A mathematical expression that involves variables and constants.
Equation: A mathematical statement that asserts the equality of two expressions.
Variable: A symbol that represents a value that can change.
Constant: A value that does not change.
Polynomial: A mathematical expression that involves variables and constants, where the variables are raised to non-negative integer powers.
Transcendental equation: A mathematical equation that involves a transcendental function, such as the exponential function or the trigonometric function.

Introduction

In our previous article, we solved a mathematical problem that involved algebraic expressions and equations. The problem stated 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. We found that the solution to the problem is x = 7. In this article, we will answer some frequently asked questions related to this problem.

Q&A

Q: What is the equation that we solved in the previous article?

A: The equation that we solved is x^2 + (x - 1) = (x + 1)^2 - 9.

Q: How did we simplify the equation?

A: We simplified the equation by expanding the right-hand side of the equation and combining like terms.

Q: What is the solution to the equation?

A: The solution to the equation is x = 7.

Q: Can you explain the concept of algebraic expressions and equations?

A: Algebraic expressions and equations are mathematical statements that involve variables and constants. Variables are symbols that represent values that can change, while constants are values that do not change. Equations are mathematical statements that assert the equality of two expressions.

Q: What is the difference between a polynomial and a transcendental equation?

A: A polynomial is a mathematical expression that involves variables and constants, where the variables are raised to non-negative integer powers. A transcendental equation, on the other hand, is a mathematical equation that involves a transcendental function, such as the exponential function or the trigonometric function.

Q: How can we use algebraic expressions and equations in real-world applications?

A: Algebraic expressions and equations have numerous real-world applications, including physics, engineering, and computer science. For example, in physics, the equation of motion can be represented as a quadratic equation, which can be solved using algebraic methods. In engineering, the design of structures such as bridges and buildings requires the use of mathematical models, which can be represented as algebraic equations.

Q: Can you provide some examples of algebraic expressions and equations?

A: Here are some examples of algebraic expressions and equations:

x^2 + 2x + 1 = 0
2x - 3 = 5
x + 2 = 7

Q: How can we solve algebraic expressions and equations?

A: There are several methods for solving algebraic expressions and equations, including factoring, the quadratic formula, and substitution.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: Can you explain the concept of substitution?

A: Substitution is a method for solving algebraic expressions and equations by substituting one expression for another. For example, if we have the equation x + 2 = 7, we can substitute x = 5 into the equation to get 5 + 2 = 7.

Conclusion

In this article, we have answered some frequently asked questions related to the mathematical problem that we solved in our previous article. We have explained the concept of algebraic expressions and equations, provided examples of algebraic expressions and equations, and discussed some methods for solving them. We hope that this article has been helpful in clarifying any questions or doubts that you may have had.

Glossary

Algebraic expression: A mathematical expression that involves variables and constants.
Equation: A mathematical statement that asserts the equality of two expressions.
Variable: A symbol that represents a value that can change.
Constant: A value that does not change.
Polynomial: A mathematical expression that involves variables and constants, where the variables are raised to non-negative integer powers.
Transcendental equation: A mathematical equation that involves a transcendental function, such as the exponential function or the trigonometric function.
Quadratic formula: A mathematical formula that is used to solve quadratic equations.
Substitution: A method for solving algebraic expressions and equations by substituting one expression for another.