Solve The Quadratic Equation. Separate Solutions With A Comma.$(2n - 2)^2 = 169$n = \square$

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, $(2n - 2)^2 = 169$, and provide a step-by-step guide on how to find the value of $n$.

What is a Quadratic Equation?

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, and $x$ is the variable. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and graphing.

The Quadratic Equation in Question

The quadratic equation we will be solving is $(2n - 2)^2 = 169$. This equation is already in the form of a quadratic equation, with the variable $n$ and the constants $2$ and $169$. Our goal is to find the value of $n$ that satisfies this equation.

Step 1: Expand the Left Side of the Equation

To solve the equation, we need to expand the left side of the equation. We can do this by using the formula $(a-b)^2 = a^2 - 2ab + b^2$. In this case, $a = 2n$ and $b = 2$. So, we have:

$(2n - 2)^2 = (2n)^2 - 2(2n)(2) + 2^2$

Expanding the left side of the equation, we get:

$4n^2 - 8n + 4 = 169$

Step 2: Simplify the Equation

Now that we have expanded the left side of the equation, we can simplify it by combining like terms. We can do this by subtracting $169$ from both sides of the equation:

$4n^2 - 8n + 4 - 169 = 0$

Simplifying the equation, we get:

$4n^2 - 8n - 165 = 0$

Step 3: Solve for $n$

Now that we have simplified the equation, we can solve for $n$. We can do this by using the quadratic formula, which is:

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

In this case, $a = 4$, $b = -8$, and $c = -165$. Plugging these values into the quadratic formula, we get:

$n = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(4)(-165)}}{2(4)}$

Simplifying the equation, we get:

$n = \frac{8 \pm \sqrt{64 + 2640}}{8}$

$n = \frac{8 \pm \sqrt{2704}}{8}$

$n = \frac{8 \pm 52}{8}$

Step 4: Find the Two Solutions

Now that we have solved for $n$, we can find the two solutions. We can do this by plugging in the values of $n$ into the original equation:

$n = \frac{8 + 52}{8}$ or $n = \frac{8 - 52}{8}$

Simplifying the equations, we get:

$n = \frac{60}{8}$ or $n = \frac{-44}{8}$

$n = 7.5$ or $n = -5.5$

Conclusion

In this article, we solved the quadratic equation $(2n - 2)^2 = 169$ and found the value of $n$. We used the quadratic formula to solve for $n$ and found two solutions: $n = 7.5$ and $n = -5.5$. We hope this article has provided a clear and step-by-step guide on how to solve quadratic equations.

Final Answer

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Introduction

In our previous article, we solved the quadratic equation $(2n - 2)^2 = 169$ and found the value of $n$. However, we understand that quadratic equations can be complex and confusing, especially for those who are new to the subject. In this article, we will answer some frequently asked questions about quadratic equations, providing a better understanding of this mathematical concept.

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, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, the quadratic formula, and graphing. The quadratic formula is a popular method, which is:

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

Q: What is the quadratic formula?

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

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

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the quadratic equation. Then, plug these values into the formula and simplify.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. A quadratic equation has a highest power of two, while a linear equation has a highest power of one.

Q: Can I solve a quadratic equation by graphing?

A: Yes, you can solve a quadratic equation by graphing. Graphing involves plotting the quadratic equation on a coordinate plane and finding the x-intercepts, which represent the solutions to the equation.

Q: What are the applications of quadratic equations?

A: Quadratic equations have numerous applications in various fields, including physics, engineering, economics, and computer science. They are used to model real-world problems, such as projectile motion, optimization problems, and data analysis.

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in quadratic formula function that can be used to solve quadratic equations.

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

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

• Not identifying the values of $a$, $b$, and $c$ correctly
• Not simplifying the equation correctly
• Not using the correct formula or method
• Not checking the solutions for validity

Conclusion

In this article, we answered some frequently asked questions about quadratic equations, providing a better understanding of this mathematical concept. We hope this article has been helpful in clarifying any doubts or questions you may have had about quadratic equations.

Final Answer

The final answer is: There is no final answer, as this article is a Q&A session. However, we hope that the information provided has been helpful in understanding quadratic equations.