How Many And What Type Of Solutions Does $6x^2 - 2x + 7 = 0$ Have?A. 2 Irrational Solutions B. 2 Nonreal Solutions C. 2 Rational Solutions D. 1 Rational Solution

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Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. In this article, we will focus on solving the quadratic equation $6x^2 - 2x + 7 = 0$ and determine the number and type of solutions it has.

Understanding Quadratic Equations

A quadratic equation is generally written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. The solutions to a quadratic equation are the values of $x$ that satisfy the equation. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

This formula provides two solutions for the quadratic equation, which are the values of $x$ that satisfy the equation.

Applying the Quadratic Formula to the Given Equation

Now, let's apply the quadratic formula to the given equation $6x^2 - 2x + 7 = 0$. We have $a = 6$, $b = -2$, and $c = 7$. Plugging these values into the quadratic formula, we get:

x=−(−2)±(−2)2−4(6)(7)2(6)x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(6)(7)}}{2(6)}

Simplifying the expression, we get:

x=2±4−16812x = \frac{2 \pm \sqrt{4 - 168}}{12}

x=2±−16412x = \frac{2 \pm \sqrt{-164}}{12}

Analyzing the Solutions

The solutions to the quadratic equation are given by the expression $x = \frac{2 \pm \sqrt{-164}}{12}$. We can see that the expression under the square root is negative, which means that the solutions will be complex numbers.

Conclusion

In conclusion, the quadratic equation $6x^2 - 2x + 7 = 0$ has two nonreal solutions. The solutions are complex numbers, which are given by the expression $x = \frac{2 \pm \sqrt{-164}}{12}$. This means that the correct answer is B. 2 nonreal solutions.

Final Answer

The final answer is B. 2 nonreal solutions.

Additional Information

It's worth noting that the solutions to the quadratic equation can be expressed in the form $x = a \pm bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit. In this case, the solutions are $x = \frac{1}{6} \pm \frac{\sqrt{41}}{6}i$.

Real-World Applications

Quadratic equations have numerous real-world applications, including physics, engineering, and economics. For example, the motion of an object under the influence of gravity can be modeled using quadratic equations. Additionally, quadratic equations are used in finance to calculate the value of investments and in computer science to optimize algorithms.

Common Mistakes to Avoid

When solving quadratic equations, it's essential to avoid common mistakes, such as:

  • Not checking the discriminant ($b^2 - 4ac$) to determine the nature of the solutions.
  • Not simplifying the expression under the square root.
  • Not expressing the solutions in the correct form ($x = a \pm bi$).

Tips and Tricks

Here are some tips and tricks to help you solve quadratic equations:

  • Use the quadratic formula to find the solutions.
  • Simplify the expression under the square root.
  • Express the solutions in the correct form ($x = a \pm bi$).
  • Check the discriminant to determine the nature of the solutions.

Conclusion

In conclusion, solving quadratic equations is a crucial skill in mathematics, and it has numerous real-world applications. By understanding the quadratic formula and applying it correctly, you can find the solutions to quadratic equations. Remember to avoid common mistakes and follow the tips and tricks provided in this article.

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Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and economics. In our previous article, we discussed how to solve quadratic equations using the quadratic formula and determined the number and type of solutions for the equation $6x^2 - 2x + 7 = 0$. In this article, we will provide a Q&A guide to help you better understand quadratic equations and how to solve them.

Q&A: 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 is two. It is generally written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. The quadratic formula is a powerful tool for solving quadratic equations and is given by:

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

Q: What is the discriminant, and why is it important?

A: The discriminant is the expression under the square root in the quadratic formula, which is given by $b^2 - 4ac$. It is essential to check the discriminant to determine the nature of the solutions. If the discriminant is positive, the solutions are real and distinct. If the discriminant is zero, the solutions are real and equal. If the discriminant is negative, the solutions are complex.

Q: How do I determine the number and type of solutions for a quadratic equation?

A: To determine the number and type of solutions for a quadratic equation, you need to check the discriminant. If the discriminant is positive, the equation has two real and distinct solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.

Q: What is the difference between a rational and irrational solution?

A: A rational solution is a solution that can be expressed as a ratio of two integers, while an irrational solution is a solution that cannot be expressed as a ratio of two integers.

Q: How do I express complex solutions in the correct form?

A: Complex solutions can be expressed in the form $x = a \pm bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

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 the discriminant to determine the nature of the solutions.
  • Not simplifying the expression under the square root.
  • Not expressing the solutions in the correct form ($x = a \pm bi$).

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have numerous real-world applications. By understanding the quadratic formula and applying it correctly, you can find the solutions to quadratic equations. Remember to avoid common mistakes and follow the tips and tricks provided in this article.

Final Tips and Tricks

Here are some final tips and tricks to help you solve quadratic equations:

  • Use the quadratic formula to find the solutions.
  • Simplify the expression under the square root.
  • Express the solutions in the correct form ($x = a \pm bi$).
  • Check the discriminant to determine the nature of the solutions.

Additional Resources

For more information on quadratic equations, we recommend the following resources:

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

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have numerous real-world applications. By understanding the quadratic formula and applying it correctly, you can find the solutions to quadratic equations. Remember to avoid common mistakes and follow the tips and tricks provided in this article.