Given { (x-7)^2 = 36$}$, Select The Values Of { X$}$.A. { X = 13$}$ B. { X = 1$}$ C. { X = -29$}$ D. { X = 42$}$

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 quadratic equations of the form ${(x-a)^2 = b}$, where ${a}$ and ${b}$ are constants. We will use the given equation ${(x-7)^2 = 36}$ as an example to demonstrate the steps involved in solving quadratic equations.

Understanding the Equation

The given equation is a quadratic equation in the form ${(x-a)^2 = b}$. In this case, ${a = 7}$ and ${b = 36}$. To solve for ${x}$, we need to isolate the variable ${x}$ on one side of the equation.

Step 1: Take the Square Root

The first step in solving the equation is to take the square root of both sides. This will help us to eliminate the squared term and get closer to isolating the variable ${x}$.

${(x-7)^2 = 36}$

Taking the square root of both sides gives us:

${\sqrt{(x-7)^2} = \sqrt{36}}$

Simplifying the left-hand side, we get:

${|x-7| = 6}$

Step 2: Simplify the Absolute Value

The absolute value equation ${|x-7| = 6}$ can be simplified by considering two cases: ${x-7 = 6}$ and ${x-7 = -6}$.

Case 1: ${x-7 = 6}$

Solving for ${x}$ in the equation ${x-7 = 6}$, we get:

${x = 6 + 7}$

${x = 13}$

Case 2: ${x-7 = -6}$

Solving for ${x}$ in the equation ${x-7 = -6}$, we get:

${x = -6 + 7}$

${x = 1}$

Step 3: Check the Solutions

Now that we have found two possible solutions for ${x}$, we need to check if they satisfy the original equation.

Substituting ${x = 13}$ into the original equation, we get:

${(13-7)^2 = 36}$

${6^2 = 36}$

${36 = 36}$

This confirms that ${x = 13}$ is a valid solution.

Substituting ${x = 1}$ into the original equation, we get:

${(1-7)^2 = 36}$

${(-6)^2 = 36}$

${36 = 36}$

This confirms that ${x = 1}$ is also a valid solution.

Conclusion

In this article, we have demonstrated the steps involved in solving quadratic equations of the form ${(x-a)^2 = b}$. We used the given equation ${(x-7)^2 = 36}$ as an example and found two possible solutions for ${x}$: ${x = 13}$ and ${x = 1}$. We also checked these solutions to confirm that they satisfy the original equation.

Final Answer

The final answer is:

• A. ${x = 13}$
• B. ${x = 1}$
• C. ${x = -29}$ (This is not a valid solution)
• D. ${x = 42}$ (This is not a valid solution)

Frequently Asked Questions

In this article, we will address some of the most common questions related to quadratic equations. Whether you are a student or a professional, this Q&A section will provide you with a better understanding of quadratic equations and how to solve them.

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 (usually x) is two. The general form of a quadratic equation is ${ax^2 + bx + c = 0}$, where ${a}$, ${b}$, and ${c}$ are constants.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following steps:

1. Take the square root: Take the square root of both sides of the equation to eliminate the squared term.
2. Simplify the absolute value: Simplify the absolute value equation to get two possible solutions.
3. Check the solutions: Check each solution to confirm that it satisfies the original equation.

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

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable (usually x) is one. The general form of a linear equation is ${ax + b = 0}$, where ${a}$ and ${b}$ are constants. Quadratic equations, on the other hand, have a squared term, which makes them more complex to solve.

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

A: Yes, you can use a calculator to solve quadratic equations. Most calculators have a built-in quadratic formula function that can help you solve quadratic equations quickly and accurately.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that can be used to solve quadratic equations. The formula is:

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

This formula can be used to find the solutions to a quadratic equation.

Q: Can I use the quadratic formula to solve all types of quadratic equations?

A: Yes, the quadratic formula can be used to solve all types of quadratic equations, including those with complex solutions.

Q: What is the difference between a quadratic equation with real solutions and one with complex solutions?

A: A quadratic equation with real solutions has two distinct real roots, while a quadratic equation with complex solutions has two complex roots.

Q: Can I use the quadratic formula to solve quadratic equations with complex solutions?

A: Yes, the quadratic formula can be used to solve quadratic equations with complex solutions.

Conclusion

In this Q&A article, we have addressed some of the most common questions related to quadratic equations. Whether you are a student or a professional, this article will provide you with a better understanding of quadratic equations and how to solve them.

Final Tips

• Always check your solutions to confirm that they satisfy the original equation.
• Use the quadratic formula to solve quadratic equations quickly and accurately.
• Practice solving quadratic equations to become more comfortable with the process.

Common Quadratic Equations

Here are some common quadratic equations that you may encounter:

• ${x^2 + 5x + 6 = 0}$
• ${x^2 - 7x + 12 = 0}$
• ${x^2 + 2x - 15 = 0}$

Solving Quadratic Equations with Complex Solutions

Here are some examples of quadratic equations with complex solutions:

• ${x^2 + 1 = 0}$
• ${x^2 - 4 = 0}$
• ${x^2 + 3x + 2 = 0}$

Conclusion

In this article, we have provided you with a comprehensive guide to quadratic equations, including how to solve them and some common examples. Whether you are a student or a professional, this article will provide you with a better understanding of quadratic equations and how to solve them.