Select The Correct Answer.Solve The Following Equation For $x$:$x^2 - 36 = 0$A. \$x = 1 ; X = -36$[/tex\] B. $x = -1 ; X = 36$ C. $x = -6 ; X = 6$ D. \$x = -18 ; X = 18$[/tex\]

Feb 28, 2025 by ADMIN 192 views

Solving Quadratic Equations: A Step-by-Step Guide

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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 $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants. We will use the given equation $x^2 - 36 = 0$ as an example to demonstrate the steps involved in solving quadratic equations.

Understanding the Equation

The given equation is $x^2 - 36 = 0$ . To solve this equation, we need to find the values of $x$ that satisfy the equation. The equation can be rewritten as $x^2 = 36$ , which is a quadratic equation in the form of $ax^2 = c$ .

Factoring the Equation

To solve the equation $x^2 = 36$ , we can factor the left-hand side as $(x + 6)(x - 6) = 0$ . This is a difference of squares, which can be factored into two binomials.

Applying the Zero Product Property

The zero product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. In this case, we have $(x + 6)(x - 6) = 0$ , which means that either $(x + 6) = 0$ or $(x - 6) = 0$ .

Solving for x

To solve for $x$ , we can set each factor equal to zero and solve for $x$ . For the first factor, we have $x + 6 = 0$ , which gives us $x = -6$ . For the second factor, we have $x - 6 = 0$ , which gives us $x = 6$ .

Checking the Solutions

To check our solutions, we can plug them back into the original equation. For $x = -6$ , we have $(-6)^2 - 36 = 36 - 36 = 0$ , which is true. For $x = 6$ , we have $(6)^2 - 36 = 36 - 36 = 0$ , which is also true.

Conclusion

In conclusion, the solutions to the equation $x^2 - 36 = 0$ are $x = -6$ and $x = 6$ . These solutions satisfy the equation and can be verified by plugging them back into the original equation.

Answer

The correct answer is:

C. $x = -6 ; x = 6$

This answer is based on the fact that the solutions to the equation $x^2 - 36 = 0$ are $x = -6$ and $x = 6$ , which are the values of $x$ that satisfy the equation.

Discussion

The equation $x^2 - 36 = 0$ is a quadratic equation that can be solved using factoring and the zero product property. The solutions to the equation are $x = -6$ and $x = 6$ , which can be verified by plugging them back into the original equation. This equation is a classic example of a quadratic equation that can be solved using factoring, and it is an important concept in mathematics.

Example Use Cases

Quadratic equations have many real-world applications, including:

Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
Computer Science: Quadratic equations are used in algorithms and data structures, such as sorting and searching.

Tips and Tricks

Here are some tips and tricks for solving quadratic equations:

Use factoring: Factoring is a powerful technique for solving quadratic equations.
Use the zero product property: The zero product property is a useful tool for solving quadratic equations.
Check your solutions: Always check your solutions by plugging them back into the original equation.

Conclusion

In conclusion, solving quadratic equations is a crucial skill for students and professionals alike. By understanding the equation, factoring, applying the zero product property, solving for x, checking the solutions, and using example use cases, we can solve quadratic equations with ease. With practice and patience, anyone can become proficient in solving quadratic equations.

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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 answer some of the most frequently asked questions about quadratic equations.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means that 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: There are several methods for solving quadratic equations, including factoring, the quadratic formula, and graphing. The method you choose will depend on the specific equation and your personal preference.

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 $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$ , $b$ , and $c$ into the formula. You will then get two solutions for $x$ , which can be found using the plus-or-minus sign.

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. In other words, a quadratic equation has a squared variable, while a linear equation does not.

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

A: No, the quadratic formula can only be used to solve quadratic equations that can be written in the form $ax^2 + bx + c = 0$ . If the equation is not in this form, you may need to use a different method to solve it.

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

A: To check your solutions, you need to plug them back into the original equation and see if they are true. If they are true, then you have found the correct solutions.

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

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

Not checking your solutions: Always check your solutions by plugging them back into the original equation.
Not using the correct method: Make sure you are using the correct method for solving the quadratic equation.
Not simplifying your solutions: Simplify your solutions as much as possible to make them easier to work with.

Q: How do I apply quadratic equations in real-world situations?

A: Quadratic equations have many real-world applications, including:

Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
Computer Science: Quadratic equations are used in algorithms and data structures, such as sorting and searching.

Q: Can I use quadratic equations to solve problems in other areas of mathematics?

A: Yes, quadratic equations can be used to solve problems in other areas of mathematics, including algebra, geometry, and trigonometry.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By understanding the basics of quadratic equations, including the quadratic formula and how to apply them in real-world situations, you can solve quadratic equations with ease. With practice and patience, anyone can become proficient in solving quadratic equations.

Additional Resources

For more information on quadratic equations, including tutorials, examples, and practice problems, check out the following resources:

Mathway: A math problem solver that can help you solve quadratic equations and other math problems.
Khan Academy: A free online resource that offers video tutorials and practice problems on quadratic equations and other math topics.
Wolfram Alpha: A powerful online calculator that can help you solve quadratic equations and other math problems.