Solve For $x$.$3x^2 - 6x + 2 = 0$A. $x = \frac{3 \pm \sqrt{3}}{3}$B. $x = \frac{6 \pm 2\sqrt{3}}{3}$C. $x = \frac{6 \pm 4\sqrt{3}}{3}$D. $x = \frac{3 \pm 2\sqrt{3}}{3}$

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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 a specific quadratic equation, 3x2−6x+2=03x^2 - 6x + 2 = 0, and explore the different methods and techniques used to find the solutions.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, xx) is two. The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants. In our equation, a=3a = 3, b=−6b = -6, and c=2c = 2.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form ax2+bx+c=0ax^2 + bx + c = 0, the solutions are given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In our equation, a=3a = 3, b=−6b = -6, and c=2c = 2. Plugging these values into the quadratic formula, we get:

x=−(−6)±(−6)2−4(3)(2)2(3)x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(3)(2)}}{2(3)}

Simplifying the Quadratic Formula

Let's simplify the expression under the square root:

(−6)2−4(3)(2)=36−24=12(-6)^2 - 4(3)(2) = 36 - 24 = 12

Now, we can rewrite the quadratic formula as:

x=6±126x = \frac{6 \pm \sqrt{12}}{6}

Rationalizing the Denominator

To rationalize the denominator, we can multiply both the numerator and the denominator by 12\sqrt{12}:

x=6±126⋅1212x = \frac{6 \pm \sqrt{12}}{6} \cdot \frac{\sqrt{12}}{\sqrt{12}}

This simplifies to:

x=6±236x = \frac{6 \pm 2\sqrt{3}}{6}

Simplifying Further

We can simplify the expression further by dividing both the numerator and the denominator by 6:

x=6±236=3±33x = \frac{6 \pm 2\sqrt{3}}{6} = \frac{3 \pm \sqrt{3}}{3}

Conclusion

In this article, we have solved the quadratic equation 3x2−6x+2=03x^2 - 6x + 2 = 0 using the quadratic formula. We have also simplified the expression to find the final solutions. The solutions to the equation are:

x=3±33x = \frac{3 \pm \sqrt{3}}{3}

This is the correct answer, and it matches option A.

Discussion

The quadratic formula is a powerful tool for solving quadratic equations. It can be used to find the solutions to any quadratic equation, as long as the coefficients are known. In this article, we have used the quadratic formula to solve the equation 3x2−6x+2=03x^2 - 6x + 2 = 0. We have also simplified the expression to find the final solutions.

Common Mistakes

When solving quadratic equations, there are several common mistakes to avoid. These include:

  • Not simplifying the expression under the square root: This can lead to incorrect solutions.
  • Not rationalizing the denominator: This can also lead to incorrect solutions.
  • Not simplifying the final expression: This can make it difficult to read and understand the solutions.

Tips and Tricks

When solving quadratic equations, there are several tips and tricks to keep in mind. These include:

  • Using the quadratic formula: This is a powerful tool for solving quadratic equations.
  • Simplifying the expression under the square root: This can make it easier to find the solutions.
  • Rationalizing the denominator: This can also make it easier to find the solutions.
  • Simplifying the final expression: This can make it easier to read and understand the solutions.

Real-World Applications

Quadratic equations have many real-world applications. These include:

  • 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.
  • Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Conclusion

In this article, we have solved the quadratic equation 3x2−6x+2=03x^2 - 6x + 2 = 0 using the quadratic formula. We have also simplified the expression to find the final solutions. The solutions to the equation are:

x=3±33x = \frac{3 \pm \sqrt{3}}{3}

This is the correct answer, and it matches option A.

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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 provide a Q&A guide to help you understand and solve quadratic equations.

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 (in this case, xx) is two. The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc 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 ax2+bx+c=0ax^2 + bx + c = 0, the solutions are given by:

x=−b±b2−4ac2ax = \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 aa, bb, and cc into the formula. Then, simplify the expression under the square root and rationalize the denominator.

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. The quadratic formula is a general method that can be used to solve any quadratic equation, while factoring is a specific method that can be used to solve quadratic equations that can be factored.

Q: When should I use the quadratic formula?

A: You should use the quadratic formula when:

  • The equation cannot be factored.
  • The equation has complex solutions.
  • You need to find the solutions to a quadratic equation quickly.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula include:

  • Not simplifying the expression under the square root.
  • Not rationalizing the denominator.
  • Not simplifying the final expression.

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

A: To simplify the expression under the square root, you need to:

  • Combine like terms.
  • Factor out any common factors.
  • Simplify any complex expressions.

Q: How do I rationalize the denominator?

A: To rationalize the denominator, you need to:

  • Multiply both the numerator and the denominator by the conjugate of the denominator.
  • Simplify the expression.

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

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.
  • Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Q: How do I know if a quadratic equation has real or complex solutions?

A: To determine if a quadratic equation has real or complex solutions, you need to:

  • Check the discriminant (b2−4acb^2 - 4ac).
  • If the discriminant is positive, the equation has two real solutions.
  • If the discriminant is zero, the equation has one real solution.
  • If the discriminant is negative, the equation has two complex solutions.

Conclusion

In this article, we have provided a Q&A guide to help you understand and solve quadratic equations. We have covered topics such as the quadratic formula, factoring, and real-world applications of quadratic equations. By following the tips and tricks outlined in this article, you will be able to solve quadratic equations with ease.