6 + 2x^2 - 3x = 8x^2

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Introduction

In mathematics, equations are a fundamental concept that help us understand and describe various relationships between variables. One of the most common types of equations is the quadratic equation, which involves a polynomial of degree two. In this article, we will focus on solving the equation 6 + 2x^2 - 3x = 8x^2, which is a quadratic equation. We will break down the solution into step-by-step instructions, making it easy to understand and follow.

Understanding the Equation

Before we dive into solving the equation, let's first understand what it represents. The equation 6 + 2x^2 - 3x = 8x^2 is a quadratic equation in the variable x. The left-hand side of the equation consists of a constant term (6), a quadratic term (2x^2), and a linear term (-3x). The right-hand side of the equation is a quadratic term (8x^2).

Rearranging the Equation

To solve the equation, we need to isolate the variable x. The first step is to rearrange the equation by moving all the terms to one side of the equation. We can do this by subtracting 8x^2 from both sides of the equation:

6 + 2x^2 - 3x - 8x^2 = 0

This simplifies to:

-6x^2 - 3x + 6 = 0

Combining Like Terms

The next step is to combine like terms. In this case, we have two quadratic terms (-6x^2 and 2x^2) and a linear term (-3x). We can combine the quadratic terms by adding their coefficients:

-6x^2 + 2x^2 = -4x^2

So, the equation becomes:

-4x^2 - 3x + 6 = 0

Factoring the Equation

Now that we have combined like terms, we can try to factor the equation. Factoring an equation involves expressing it as a product of two or more binomials. In this case, we can factor the equation as follows:

-4x^2 - 3x + 6 = (-2x + 3)(2x - 2) = 0

Solving for x

Now that we have factored the equation, we can solve for x. To do this, we need to set each factor equal to zero and solve for x:

-2x + 3 = 0 --> x = 3/2

2x - 2 = 0 --> x = 1

Conclusion

In this article, we have solved the equation 6 + 2x^2 - 3x = 8x^2 using step-by-step instructions. We started by rearranging the equation, combining like terms, factoring the equation, and finally solving for x. The solutions to the equation are x = 3/2 and x = 1. We hope that this article has provided a clear and concise guide to solving quadratic equations.

Additional Tips and Tricks

  • When solving quadratic equations, it's essential to check your solutions by plugging them back into the original equation.
  • If the equation has no real solutions, it may have complex solutions. In this case, you can use the quadratic formula to find the complex solutions.
  • When factoring quadratic equations, it's often helpful to look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Frequently Asked Questions

  • Q: What is a quadratic equation? A: A quadratic equation is a polynomial equation of degree two, which means it has a highest power of two.
  • Q: How do I solve a quadratic equation? A: To solve a quadratic equation, you can use the quadratic formula, factoring, or completing the square.
  • Q: What is the quadratic formula? A: The quadratic formula is a formula that gives the solutions to a quadratic equation in the form ax^2 + bx + c = 0. The formula is x = (-b ± √(b^2 - 4ac)) / 2a.

Final Thoughts

Solving quadratic equations is an essential skill in mathematics, and it has many real-world applications. In this article, we have provided a step-by-step guide to solving the equation 6 + 2x^2 - 3x = 8x^2. We hope that this article has provided a clear and concise guide to solving quadratic equations.

Introduction

In our previous article, we solved the equation 6 + 2x^2 - 3x = 8x^2 using step-by-step instructions. However, we know that solving quadratic equations can be a challenging task, and many students struggle to understand the concepts and techniques involved. In this article, we will provide a Q&A guide to help you better understand and solve quadratic equations.

Q&A Guide

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means it has a highest power of two. It is typically written in the form 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, completing the square, and using the quadratic formula. 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 formula that gives the solutions to a quadratic equation in the form ax^2 + bx + c = 0. The formula is x = (-b ± √(b^2 - 4ac)) / 2a.

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 it has a highest power of one. It is typically written in the form ax + b = 0, where a and b are constants. A quadratic equation, on the other hand, is a polynomial equation of degree two, which means it has a highest power of two.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions. This is because the graph of a quadratic equation is a parabola, which has a maximum or minimum point. The solutions to the equation are the x-coordinates of this point.

Q: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, you can use the discriminant, which is the expression b^2 - 4ac. If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Q: What is the discriminant?

A: The discriminant is the expression b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation. It is used to determine the number of solutions to the equation.

Q: Can a quadratic equation have complex solutions?

A: Yes, a quadratic equation can have complex solutions. If the discriminant is negative, the equation has no real solutions, but it may have complex solutions.

Q: How do I find the complex solutions to a quadratic equation?

A: To find the complex solutions to a quadratic equation, you can use the quadratic formula and take the square root of the negative discriminant.

Q: What is the difference between a quadratic equation and a polynomial equation of degree three?

A: A polynomial equation of degree three is a polynomial equation with a highest power of three. It is typically written in the form ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants. A quadratic equation, on the other hand, is a polynomial equation of degree two.

Conclusion

In this article, we have provided a Q&A guide to help you better understand and solve quadratic equations. We have covered topics such as the definition of a quadratic equation, the quadratic formula, and the discriminant. We hope that this article has provided a clear and concise guide to solving quadratic equations.

Additional Tips and Tricks

  • When solving quadratic equations, it's essential to check your solutions by plugging them back into the original equation.
  • If the equation has no real solutions, it may have complex solutions. In this case, you can use the quadratic formula to find the complex solutions.
  • When factoring quadratic equations, it's often helpful to look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Frequently Asked Questions

  • Q: What is a quadratic equation? A: A quadratic equation is a polynomial equation of degree two, which means it has a highest power of two.
  • Q: How do I solve a quadratic equation? A: To solve a quadratic equation, you can use the quadratic formula, factoring, or completing the square.
  • Q: What is the quadratic formula? A: The quadratic formula is a formula that gives the solutions to a quadratic equation in the form ax^2 + bx + c = 0. The formula is x = (-b ± √(b^2 - 4ac)) / 2a.

Final Thoughts

Solving quadratic equations is an essential skill in mathematics, and it has many real-world applications. In this article, we have provided a Q&A guide to help you better understand and solve quadratic equations. We hope that this article has provided a clear and concise guide to solving quadratic equations.