Expand the squared binomial expression

Oct 13, 2025 by magicalceylontours.com 93 views

Introduction

Solving equations involving exponents and powers can be a challenging task, especially when dealing with quadratic expressions. In this article, we will focus on solving the equation $(2x - 1)^2 = 121$ , which involves a squared binomial expression. We will use algebraic techniques to simplify the equation and find the values of $x$ that satisfy the equation.

Understanding the Equation

The given equation is $(2x - 1)^2 = 121$ . This equation involves a squared binomial expression, which can be expanded using the formula $(a - b)^2 = a^2 - 2ab + b^2$ . In this case, we have $(2x - 1)^2$ , which can be expanded as follows:

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

Simplifying the expression, we get:

(2x - 1)^2 = 4x^2 - 4x + 1

Setting Up the Equation

Now that we have expanded the squared binomial expression, we can set up the equation by equating it to the given value of 121:

4x^2 - 4x + 1 = 121

Simplifying the Equation

To simplify the equation, we can start by subtracting 121 from both sides of the equation:

4x^2 - 4x - 120 = 0

Factoring the Quadratic Expression

The resulting quadratic expression can be factored as follows:

(2x + 6)(2x - 20) = 0

Solving for $x$

To solve for $x$ , we can set each factor equal to zero and solve for $x$ :

2x + 6 = 0 \Rightarrow 2x = -6 \Rightarrow x = -3

2x - 20 = 0 \Rightarrow 2x = 20 \Rightarrow x = 10

Conclusion

In this article, we have solved the equation $(2x - 1)^2 = 121$ using algebraic techniques. We expanded the squared binomial expression, set up the equation, simplified it, factored the quadratic expression, and solved for $x$ . The solutions to the equation are $x = -3$ and $x = 10$ . These values of $x$ satisfy the equation and can be verified by plugging them back into the original equation.

Additional Tips and Tricks

When solving equations involving exponents and powers, it's essential to remember the following tips and tricks:

Expand the squared binomial expression: When dealing with squared binomial expressions, it's crucial to expand them using the formula $(a - b)^2 = a^2 - 2ab + b^2$ .
Simplify the equation: Simplifying the equation can help to make it easier to solve and can also help to identify any potential errors.
Factor the quadratic expression: Factoring the quadratic expression can help to identify the solutions to the equation and can also help to simplify the equation.
Check the solutions: It's essential to check the solutions by plugging them back into the original equation to ensure that they satisfy the equation.

Real-World Applications

Solving equations involving exponents and powers has numerous real-world applications in various fields, including:

Physics: Solving equations involving exponents and powers is essential in physics to describe the motion of objects and to calculate quantities such as velocity and acceleration.
Engineering: Solving equations involving exponents and powers is essential in engineering to design and optimize systems, such as electrical circuits and mechanical systems.
Computer Science: Solving equations involving exponents and powers is essential in computer science to develop algorithms and to solve problems in fields such as machine learning and data analysis.

Final Thoughts

Solving equations involving exponents and powers can be a challenging task, but with practice and patience, it can become a manageable and even enjoyable task. By following the tips and tricks outlined in this article, you can develop the skills and confidence needed to solve equations involving exponents and powers. Whether you're a student, a teacher, or a professional, solving equations involving exponents and powers is an essential skill that can help you to succeed in a wide range of fields.

Introduction

In our previous article, we solved the equation $(2x - 1)^2 = 121$ using algebraic techniques. In this article, we will answer some of the most frequently asked questions about solving this equation.

Q: What is the first step in solving the equation $(2x - 1)^2 = 121$ ?

A: The first step in solving the equation $(2x - 1)^2 = 121$ is to expand the squared binomial expression using the formula $(a - b)^2 = a^2 - 2ab + b^2$ . This will give us a quadratic expression that we can simplify and solve.

Q: How do I simplify the equation $(2x - 1)^2 = 121$ ?

A: To simplify the equation $(2x - 1)^2 = 121$ , we can start by subtracting 121 from both sides of the equation. This will give us a quadratic expression that we can factor and solve.

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

A: A quadratic expression is an algebraic expression that contains a squared variable, such as $x^2$ . A quadratic equation is a quadratic expression that is set equal to zero, such as $x^2 = 0$ .

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, we can look for two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the squared variable. For example, if we have the quadratic expression $x^2 + 5x + 6$ , we can factor it as $(x + 3)(x + 2)$ .

Q: What is the relationship between the solutions to a quadratic equation and the graph of the corresponding quadratic function?

A: The solutions to a quadratic equation are the x-coordinates of the points where the graph of the corresponding quadratic function intersects the x-axis. In other words, the solutions to a quadratic equation are the values of x that make the quadratic function equal to zero.

Q: How do I check my solutions to a quadratic equation?

A: To check your solutions to a quadratic equation, you can plug each solution back into the original equation and verify that it is true. This will help you to ensure that you have found all of the solutions to the equation.

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

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

Not expanding the squared binomial expression: Failing to expand the squared binomial expression can lead to incorrect solutions.
Not simplifying the equation: Failing to simplify the equation can make it difficult to solve.
Not factoring the quadratic expression: Failing to factor the quadratic expression can make it difficult to solve.
Not checking the solutions: Failing to check the solutions can lead to incorrect solutions.

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

A: Solving quadratic equations has numerous real-world applications in various fields, including:

Physics: Solving quadratic equations is essential in physics to describe the motion of objects and to calculate quantities such as velocity and acceleration.
Engineering: Solving quadratic equations is essential in engineering to design and optimize systems, such as electrical circuits and mechanical systems.
Computer Science: Solving quadratic equations is essential in computer science to develop algorithms and to solve problems in fields such as machine learning and data analysis.

Q: How can I practice solving quadratic equations?

A: There are many ways to practice solving quadratic equations, including:

Using online resources: There are many online resources available that provide practice problems and solutions for quadratic equations.
Working with a tutor: Working with a tutor can provide one-on-one instruction and feedback on solving quadratic equations.
Joining a study group: Joining a study group can provide a collaborative environment to practice solving quadratic equations.
Using a textbook: Using a textbook can provide a comprehensive guide to solving quadratic equations.

Conclusion

Solving quadratic equations is an essential skill in mathematics and has numerous real-world applications. By following the tips and tricks outlined in this article, you can develop the skills and confidence needed to solve quadratic equations. Whether you're a student, a teacher, or a professional, solving quadratic equations is an essential skill that can help you to succeed in a wide range of fields.