Solve The Equation For \[$ X \$\]:$\[ X^2 = \frac{1}{2} X + 5 \\]

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 the equation $x^2 = \frac{1}{2}x + 5$. We will break down the solution into manageable steps, using algebraic manipulations and mathematical concepts to arrive at the final answer.

Understanding the Equation

The given equation is a quadratic equation in the form of $ax^2 + bx + c = 0$, where $a = 1$, $b = -\frac{1}{2}$, and $c = -5$. To solve this equation, we need to isolate the variable $x$.

Step 1: Rearrange the Equation

The first step is to rearrange the equation to get all the terms on one side. We can do this by subtracting $\frac{1}{2}x$ from both sides of the equation:

$$x^2 - \frac{1}{2}x = 5$$

Step 2: Move the Constant Term

Next, we need to move the constant term to the right-hand side of the equation. We can do this by subtracting 5 from both sides of the equation:

$$x^2 - \frac{1}{2}x - 5 = 0$$

Step 3: Factor the Quadratic Expression

Now, we need to factor the quadratic expression on the left-hand side of the equation. Unfortunately, this expression does not factor easily, so we will need to use other methods to solve the equation.

Step 4: Use the Quadratic Formula

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

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

In our case, $a = 1$, $b = -\frac{1}{2}$, and $c = -5$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-(-\frac{1}{2}) \pm \sqrt{(-\frac{1}{2})^2 - 4(1)(-5)}}{2(1)}$$

Step 5: Simplify the Expression

Simplifying the expression inside the square root, we get:

$$x = \frac{\frac{1}{2} \pm \sqrt{\frac{1}{4} + 20}}{2}$$

Step 6: Simplify Further

Simplifying further, we get:

$$x = \frac{\frac{1}{2} \pm \sqrt{\frac{81}{4}}}{2}$$

Step 7: Simplify the Square Root

Simplifying the square root, we get:

$$x = \frac{\frac{1}{2} \pm \frac{9}{2}}{2}$$

Step 8: Simplify the Expression

Simplifying the expression, we get two possible solutions:

$$x = \frac{\frac{1}{2} + \frac{9}{2}}{2} = \frac{5}{2}$$

$$x = \frac{\frac{1}{2} - \frac{9}{2}}{2} = -2$$

Conclusion

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In this article, we solved the quadratic equation $x^2 = \frac{1}{2}x + 5$ using algebraic manipulations and the quadratic formula. We broke down the solution into manageable steps, using mathematical concepts to arrive at the final answer. The two possible solutions to the equation are $x = \frac{5}{2}$ and $x = -2$.

Real-World Applications

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 investment portfolios.

Tips and Tricks

• Use the Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations. It can be used to solve equations that do not factor easily.
• Check Your Work: Always check your work by plugging the solutions back into the original equation.
• Use Algebraic Manipulations: Algebraic manipulations can be used to simplify the equation and make it easier to solve.

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 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 is two. It is typically written in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, 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 powerful tool for solving quadratic equations. It states that for an equation in the form of $ax^2 + bx + c = 0$, the solutions are 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, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, simplify the expression and solve for $x$.

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula and factoring are two different methods for solving quadratic equations. Factoring involves finding two binomials whose product is the original equation, while the quadratic formula involves using a formula to find the solutions.

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

A: Yes, the quadratic formula can be used to solve all quadratic equations. However, it may not always be the easiest or most efficient method, especially for equations that can be easily factored.

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 work: Always check your work by plugging the solutions back into the original equation.
• Not using the correct formula: Make sure to use the correct formula for the specific type of equation you are solving.
• Not simplifying the expression: Simplify the expression as much as possible to make it easier to solve.

Q: How do I know which method to use when solving a quadratic equation?

A: The method you choose will depend on the specific equation and your personal preference. If the equation can be easily factored, factoring may be the best method. If the equation cannot be easily factored, the quadratic formula may be a better option.

Q: Can I use technology to solve quadratic equations?

A: Yes, technology can be used to solve quadratic equations. Many graphing calculators and computer algebra systems can solve quadratic equations and provide the solutions.

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 different methods for solving quadratic equations, including factoring and the quadratic formula, you can solve a wide range of equations and apply mathematical concepts to real-world problems.

Additional Resources

• Quadratic Formula Calculator: A calculator that can be used to solve quadratic equations using the quadratic formula.
• Factoring Calculator: A calculator that can be used to factor quadratic equations.
• Graphing Calculator: A calculator that can be used to graph quadratic equations and find the solutions.

Final Tips

• Practice, practice, practice: The more you practice solving quadratic equations, the more comfortable you will become with the different methods and techniques.
• Use technology: Technology can be a powerful tool for solving quadratic equations and providing the solutions.
• Check your work: Always check your work by plugging the solutions back into the original equation.