Solve For { X $}$ In The Equation:$ X^2 = 10x - 1 }$Choose The Correct Solution A. { X = {-5 \pm 6 \sqrt{2 } $}$ B. { X = {5 \pm 2 \sqrt{6}} $}$ C. { X = {-5 \pm 2 \sqrt{6}} $}$

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, $x^2 = 10x - 1$, and choose the correct solution from the given options.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In our case, the equation is $x^2 = 10x - 1$, which can be rewritten as $x^2 - 10x + 1 = 0$.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

In our case, $a = 1$, $b = -10$, and $c = 1$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(1)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{10 \pm \sqrt{100 - 4}}{2}$$

$$x = \frac{10 \pm \sqrt{96}}{2}$$

$$x = \frac{10 \pm 4\sqrt{6}}{2}$$

$$x = 5 \pm 2\sqrt{6}$$

Choosing the Correct Solution

Now that we have solved the equation, we need to choose the correct solution from the given options. Let's examine each option carefully:

• Option A: $x = \{-5 \pm 6 \sqrt{2}\}$
• Option B: $x = \{5 \pm 2 \sqrt{6}\}$
• Option C: $x = \{-5 \pm 2 \sqrt{6}\}$

Comparing our solution with each option, we can see that only Option B matches our solution exactly.

Conclusion

Solving quadratic equations requires a clear understanding of the quadratic formula and the ability to apply it to different equations. In this article, we solved the equation $x^2 = 10x - 1$ using the quadratic formula and chose the correct solution from the given options. By following the steps outlined in this article, you can solve quadratic equations with confidence.

Additional Tips and Resources

• To solve quadratic equations, make sure to identify the values of $a$, $b$, and $c$ and plug them into the quadratic formula.
• Simplify the expression under the square root by factoring or using the difference of squares formula.
• Choose the correct solution by comparing your solution with the given options.
• Practice solving quadratic equations with different values of $a$, $b$, and $c$ to become more confident in your skills.

Common Quadratic Equations and Their Solutions

Here are some common quadratic equations and their solutions:

• $x^2 + 4x + 4 = 0$ $\Rightarrow$ $x = \{-2 \pm 0\} = -2$
• $x^2 - 6x + 8 = 0$ $\Rightarrow$ $x = \{3 \pm 1\} = 2, 4$
• $x^2 + 2x - 6 = 0$ $\Rightarrow$ $x = \{-1 \pm 3\} = 1, -4$

Real-World Applications of Quadratic Equations

Quadratic equations have numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic equations are used to model economic systems, including supply and demand curves, and to optimize resource allocation.

Conclusion

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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 frequently asked questions about quadratic equations, including their definition, properties, and applications.

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. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

Q: How do I choose the correct solution?

A: To choose the correct solution, you need to compare your solution with the given options. Make sure to simplify the expression under the square root by factoring or using the difference of squares formula.

Q: What are some common quadratic equations and their solutions?

A: Here are some common quadratic equations and their solutions:

• $x^2 + 4x + 4 = 0$ $\Rightarrow$ $x = \{-2 \pm 0\} = -2$
• $x^2 - 6x + 8 = 0$ $\Rightarrow$ $x = \{3 \pm 1\} = 2, 4$
• $x^2 + 2x - 6 = 0$ $\Rightarrow$ $x = \{-1 \pm 3\} = 1, -4$

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

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

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic equations are used to model economic systems, including supply and demand curves, and to optimize resource allocation.

Q: How do I simplify the expression under the square root?

A: To simplify the expression under the square root, you can factor the expression or use the difference of squares formula. For example, if you have the expression $b^2 - 4ac$, you can factor it as $(b - 2a)(b + 2a)$.

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

A: Here are some common mistakes to avoid when solving quadratic equations:

• Not simplifying the expression under the square root: Make sure to simplify the expression under the square root by factoring or using the difference of squares formula.
• Not choosing the correct solution: Make sure to compare your solution with the given options and choose the correct solution.
• Not checking for extraneous solutions: Make sure to check for extraneous solutions by plugging the solutions back into the original equation.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we answered some frequently asked questions about quadratic equations, including their definition, properties, and applications. By following the steps outlined in this article, you can solve quadratic equations with confidence.

Additional Resources

• Quadratic Equation Calculator: A quadratic equation calculator can help you solve quadratic equations quickly and easily.
• Quadratic Equation Solver: A quadratic equation solver can help you solve quadratic equations step-by-step.
• Quadratic Equation Tutorial: A quadratic equation tutorial can provide you with a comprehensive overview of quadratic equations and their solutions.

Common Quadratic Equations and Their Solutions

Here are some common quadratic equations and their solutions:

• $x^2 + 4x + 4 = 0$ $\Rightarrow$ $x = \{-2 \pm 0\} = -2$
• $x^2 - 6x + 8 = 0$ $\Rightarrow$ $x = \{3 \pm 1\} = 2, 4$
• $x^2 + 2x - 6 = 0$ $\Rightarrow$ $x = \{-1 \pm 3\} = 1, -4$

Real-World Applications of Quadratic Equations

Quadratic equations have numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic equations are used to model economic systems, including supply and demand curves, and to optimize resource allocation.