Solve For { X $} . . . { 3x^2 - X = 2 \}

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 $3x^2 - x = 2$ as an example to demonstrate the steps involved in solving quadratic equations.

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. The solutions to a quadratic equation are called roots or zeros.

The Given Equation

The given equation is $3x^2 - x = 2$. To solve this equation, we need to isolate the variable $x$. The first step is to move all terms to one side of the equation, so we have:

$$3x^2 - x - 2 = 0$$

Step 1: Move All Terms to One Side

In this step, we move all terms to one side of the equation by adding or subtracting the same value to both sides. In this case, we add 2 to both sides to get:

$$3x^2 - x = 2$$

$$3x^2 - x - 2 = 0$$

Step 2: Factor the Quadratic Expression

The next step is to factor the quadratic expression on the left-hand side of the equation. Factoring involves expressing the quadratic expression as a product of two binomials. In this case, we can factor the quadratic expression as:

$$(3x + 2)(x - 1) = 0$$

Step 3: Solve for x

Now that we have factored the quadratic expression, we can solve for $x$ by setting each factor equal to zero and solving for $x$. In this case, we have:

$$(3x + 2) = 0$$

$$x - 1 = 0$$

Solving for $x$ in each equation, we get:

$$x = -\frac{2}{3}$$

$$x = 1$$

Conclusion

In this article, we have demonstrated the steps involved in solving quadratic equations. We used the given equation $3x^2 - x = 2$ as an example to illustrate the process. By following these steps, we can solve quadratic equations and find the roots or zeros of the equation.

Tips and Tricks

• When solving quadratic equations, always move all terms to one side of the equation.
• Factor the quadratic expression if possible.
• Set each factor equal to zero and solve for $x$.
• Check your solutions by plugging them back into the original equation.

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.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Common Mistakes to Avoid

• Not moving all terms to one side of the equation.
• Not factoring the quadratic expression.
• Not checking solutions by plugging them back into the original equation.

Conclusion

Introduction

Quadratic equations can be a challenging topic for many students and professionals. In this article, we will answer some of the most frequently asked questions about quadratic equations, providing you with a better understanding of this important 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. 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 need to follow these steps:

1. Move all terms to one side of the equation.
2. Factor the quadratic expression if possible.
3. Set each factor equal to zero and solve for $x$.
4. Check your solutions by plugging them back into 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 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 degree of two, making them more complex than linear equations.

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 be used to solve quadratic equations. However, it's always a good idea to understand the steps involved in solving quadratic equations, as this will help you to better understand the concept and make you a more confident problem-solver.

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 roots or zeros of a quadratic equation.

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's worth noting that the quadratic formula may not always be the most efficient way to solve a quadratic equation, especially if the equation can be factored easily.

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

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

• Not moving all terms to one side of the equation.
• Not factoring the quadratic expression.
• Not checking solutions by plugging them back into the original equation.
• Using the quadratic formula incorrectly.

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

A: To check your solutions to a quadratic equation, you need to plug the solutions back into the original equation and verify that they are true. This will help you to ensure that your solutions are correct and that you have not made any mistakes.

Conclusion

Quadratic equations can be a challenging topic, but with practice and patience, you can become proficient in solving them. By following the steps outlined in this article and avoiding common mistakes, you can become a more confident problem-solver and better understand the concept of quadratic equations.

Additional Resources

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equations
• Wolfram Alpha: Quadratic Equations

Final Tips

• Practice solving quadratic equations regularly to become more confident and proficient.
• Use online resources, such as calculators and math websites, to help you solve quadratic equations.
• Don't be afraid to ask for help if you are struggling with a quadratic equation.