Solve The Equation:4. { (x-1)^2 = 4$}$

Apr 3, 2025 by ADMIN 39 views

Solving the Equation: $(x-1)^2 = 4$

Introduction

In mathematics, equations involving quadratic expressions are a fundamental concept in algebra. The equation $(x-1)^2 = 4$ is a quadratic equation that can be solved using various methods. In this article, we will explore the steps to solve this equation and understand the concept behind it.

Understanding the Equation

The given equation is $(x-1)^2 = 4$ . This equation involves a quadratic expression $(x-1)^2$ , which is equal to 4. To solve this equation, we need to isolate the variable $x$ .

Step 1: Expand the Quadratic Expression

The first step is to expand the quadratic expression $(x-1)^2$ . Using the formula $(a-b)^2 = a^2 - 2ab + b^2$ , we can expand the expression as follows:

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

Simplifying the expression, we get:

$x^2 - 2x + 1 = 4$

Step 2: Rearrange the Equation

The next step is to rearrange the equation to isolate the variable $x$ . We can do this by subtracting 4 from both sides of the equation:

$x^2 - 2x + 1 - 4 = 0$

Simplifying the equation, we get:

$x^2 - 2x - 3 = 0$

Step 3: Solve the Quadratic Equation

Now that we have a quadratic equation in the form $ax^2 + bx + c = 0$ , we can use various methods to solve it. In this case, we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substituting the values of $a$ , $b$ , and $c$ into the formula, we get:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-3)}}{2(1)}$

Simplifying the expression, we get:

$x = \frac{2 \pm \sqrt{4 + 12}}{2}$

$x = \frac{2 \pm \sqrt{16}}{2}$

$x = \frac{2 \pm 4}{2}$

Step 4: Find the Solutions

Now that we have the solutions in the form $x = \frac{2 \pm 4}{2}$ , we can simplify them to find the final solutions:

$x = \frac{2 + 4}{2}$

$x = \frac{6}{2}$

$x = 3$

$x = \frac{2 - 4}{2}$

$x = \frac{-2}{2}$

$x = -1$

Conclusion

In this article, we solved the equation $(x-1)^2 = 4$ using various methods. We expanded the quadratic expression, rearranged the equation, and used the quadratic formula to find the solutions. The final solutions are $x = 3$ and $x = -1$ . These solutions satisfy the original equation and provide a complete understanding of the concept.

Additional Tips and Tricks

When solving quadratic equations, it's essential to check the solutions by substituting them back into the original equation.
The quadratic formula can be used to solve quadratic equations in the form $^2 + bx + c = 0$ .
When expanding quadratic expressions, it's crucial to use the correct formula and simplify the expression carefully.

Frequently Asked Questions

Q: What is the quadratic formula? A: The quadratic formula is a method used to solve quadratic equations in the form $ax^2 + bx + c = 0$ .
Q: How do I expand a quadratic expression? A: To expand a quadratic expression, use the formula $(a-b)^2 = a^2 - 2ab + b^2$ .
Q: What are the solutions to the equation $(x-1)^2 = 4$ ? A: The solutions to the equation $(x-1)^2 = 4$ are $x = 3$ and $x = -1$ .

Final Thoughts

Solving quadratic equations is a fundamental concept in algebra. By understanding the steps involved in solving these equations, we can apply this knowledge to various mathematical problems. The equation $(x-1)^2 = 4$ is a simple example of a quadratic equation, and by following the steps outlined in this article, we can find the solutions and gain a deeper understanding of the concept.

Introduction

Quadratic equations are a fundamental concept in algebra, and solving them can be a challenging task for many students. In our previous article, we explored the steps to solve the equation $(x-1)^2 = 4$ . In this article, we will provide a Q&A guide to help you understand the concept of solving quadratic equations.

Q&A Guide

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is typically written in the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I identify a quadratic equation?

A: To identify a quadratic equation, look for the following characteristics:

The equation has a highest power of two.
The equation is in the form $ax^2 + bx + c = 0$ .
The equation has a variable (usually $x$ ) and constants.

Q: What are the steps to solve a quadratic equation?

A: The steps to solve a quadratic equation are:

Expand the quadratic expression (if necessary).
Rearrange the equation to isolate the variable.
Use the quadratic formula to find the solutions.
Check the solutions by substituting them back into the original equation.

Q: What is the quadratic formula?

A: The quadratic formula is a method used to solve quadratic equations in the form $ax^2 + bx + c = 0$ . It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: How do I use the quadratic formula?

A: To use the quadratic formula, follow these steps:

Identify the values of $a$ , $b$ , and $c$ in the equation.
Plug these values into the quadratic formula.
Simplify the expression to find the solutions.

Q: What are the solutions to a quadratic equation?

A: The solutions to a quadratic equation are the values of the variable that satisfy the equation. They can be real or complex numbers.

Q: How do I check the solutions?

A: To check the solutions, substitute them back into the original equation. If the equation is true, then the solution is correct.

Q: What are the different types of solutions to a quadratic equation?

A: The solutions to a quadratic equation can be:

Real and distinct (two different real solutions).
Real and equal (one real solution).
Complex (two complex solutions).

Q: How do I determine the type of solution?

A: To determine the type of solution, use the discriminant ( $b^2 - 4ac$ ) in the quadratic formula. If the discriminant is:

Positive, the solutions are real and distinct.
Zero, the solutions are real and equal.
Negative, the solutions are complex.

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

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

Not expanding the quadratic expression correctly.
Not rearranging the equation to isolate the variable.
Not using the correct values in the quadratic formula.
Not checking the solutions.

Conclusion

Solving quadratic equations be a challenging task, but with practice and patience, you can master the concept. By following the steps outlined in this Q&A guide, you can confidently solve quadratic equations and gain a deeper understanding of the concept.

Additional Tips and Tricks

When solving quadratic equations, it's essential to check the solutions by substituting them back into the original equation.
The quadratic formula can be used to solve quadratic equations in the form $ax^2 + bx + c = 0$ .
When expanding quadratic expressions, it's crucial to use the correct formula and simplify the expression carefully.

Frequently Asked Questions

Q: What is the quadratic formula? A: The quadratic formula is a method used to solve quadratic equations in the form $ax^2 + bx + c = 0$ .
Q: How do I use the quadratic formula? A: To use the quadratic formula, follow the steps outlined in this Q&A guide.
Q: What are the solutions to a quadratic equation? A: The solutions to a quadratic equation are the values of the variable that satisfy the equation.

Final Thoughts

Solving quadratic equations is a fundamental concept in algebra, and by understanding the steps involved in solving these equations, you can apply this knowledge to various mathematical problems. The Q&A guide provided in this article will help you to confidently solve quadratic equations and gain a deeper understanding of the concept.