Use The Quadratic Formula To Find Both Solutions To The Quadratic Equation Given Below.$3x^2 - 7x - 1 = 0$A. $x = \frac{7 - \sqrt{61}}{6}$B. $x = \frac{7 + \sqrt{61}}{6}$C. $x = \frac{7 + \sqrt{37}}{6}$D. $x =

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Introduction to Quadratic Equations

Quadratic equations are a fundamental concept in algebra, and they have numerous applications in various fields such as physics, engineering, and economics. 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, and x is the variable. In this article, we will focus on solving quadratic equations using the quadratic formula.

The Quadratic Formula

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

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation. The quadratic formula provides two solutions to the quadratic equation, which are the values of x that satisfy the equation.

Step-by-Step Solution to the Quadratic Equation

Now, let's apply the quadratic formula to the given quadratic equation:

3x^2 - 7x - 1 = 0

We can identify the coefficients as follows:

a = 3, b = -7, and c = -1

Now, we can plug these values into the quadratic formula:

x = (7 ± √((-7)^2 - 4(3)(-1))) / 2(3)

x = (7 ± √(49 + 12)) / 6

x = (7 ± √61) / 6

Evaluating the Solutions

Now, we have two possible solutions to the quadratic equation:

x = (7 - √61) / 6

x = (7 + √61) / 6

We can evaluate these solutions by plugging them back into the original equation to check if they satisfy the equation.

Conclusion

In this article, we have used the quadratic formula to solve the quadratic equation 3x^2 - 7x - 1 = 0. We have identified the coefficients of the equation and plugged them into the quadratic formula to obtain two possible solutions. We have also evaluated these solutions by plugging them back into the original equation to check if they satisfy the equation.

Final Answer

The final answer to the quadratic equation 3x^2 - 7x - 1 = 0 is:

x = (7 - √61) / 6 or x = (7 + √61) / 6

Discussion

The quadratic formula is a powerful tool for solving quadratic equations. It provides two solutions to the equation, which are the values of x that satisfy the equation. In this article, we have used the quadratic formula to solve the quadratic equation 3x^2 - 7x - 1 = 0. We have identified the coefficients of the equation and plugged them into the quadratic formula to obtain two possible solutions. We have also evaluated these solutions by plugging them back into the original equation to check if they satisfy the equation.

Common Mistakes

When using the quadratic formula, there are several common mistakes that can occur. These include:

  • Incorrectly identifying the coefficients: Make sure to identify the coefficients correctly before plugging them into the quadratic formula.
  • Incorrectly evaluating the solutions: Make sure to evaluate the solutions correctly by plugging them back into the original equation to check if they satisfy the equation.
  • Not simplifying the solutions: Make sure to simplify the solutions correctly to obtain the final answer.

Real-World Applications

The quadratic formula has numerous real-world applications in various fields such as physics, engineering, and economics. Some examples include:

  • Projectile motion: The quadratic formula can be used to model the trajectory of a projectile under the influence of gravity.
  • Electrical circuits: The quadratic formula can be used to analyze the behavior of electrical circuits.
  • Economics: The quadratic formula can be used to model the behavior of economic systems.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. It provides two solutions to the equation, which are the values of x that satisfy the equation. In this article, we have used the quadratic formula to solve the quadratic equation 3x^2 - 7x - 1 = 0. We have identified the coefficients of the equation and plugged them into the quadratic formula to obtain two possible solutions. We have also evaluated these solutions by plugging them back into the original equation to check if they satisfy the equation.

Final Answer

The final answer to the quadratic equation 3x^2 - 7x - 1 = 0 is:

x = (7 - √61) / 6 or x = (7 + √61) / 6

References

  • "Quadratic Formula" by Math Open Reference. Math Open Reference, 2022.
  • "Quadratic Equations" by Khan Academy. Khan Academy, 2022.
  • "Solving Quadratic Equations" by Purplemath. Purplemath, 2022.

Glossary

  • Quadratic equation: A polynomial equation of degree two, which means the highest power of the variable is two.
  • Quadratic formula: A powerful tool for solving quadratic equations, which provides two solutions to the equation.
  • Coefficients: The constants in a quadratic equation, which are a, b, and c.
  • Solutions: The values of x that satisfy the quadratic equation.

FAQs

  • What is the quadratic formula? The quadratic formula is a powerful tool for solving quadratic equations, which provides two solutions to the equation.
  • How do I use the quadratic formula? To use the quadratic formula, you need to identify the coefficients of the quadratic equation and plug them into the formula.
  • What are the common mistakes when using the quadratic formula? The common mistakes when using the quadratic formula include incorrectly identifying the coefficients, incorrectly evaluating the solutions, and not simplifying the solutions.

Introduction

The quadratic formula is a powerful tool for solving quadratic equations. It provides two solutions to the equation, which are the values of x that satisfy the equation. In this article, we will answer some of the most frequently asked questions about the quadratic formula.

Q: What is the quadratic formula?

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

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the coefficients of the quadratic equation and plug them into the formula. The coefficients are the constants in the quadratic equation, which are a, b, and c.

Q: What are the coefficients in a quadratic equation?

A: The coefficients in a quadratic equation are the constants a, b, and c. They are the numbers that are multiplied by the variables in the equation.

Q: How do I identify the coefficients in a quadratic equation?

A: To identify the coefficients in a quadratic equation, you need to look at the equation and identify the numbers that are multiplied by the variables. For example, in the equation 3x^2 - 7x - 1 = 0, the coefficients are a = 3, b = -7, and c = -1.

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 powerful tool that provides two solutions to the equation, while factoring is a method that involves finding the factors of the quadratic expression.

Q: When should I use the quadratic formula and when should I use factoring?

A: You should use the quadratic formula when the quadratic expression cannot be factored easily, or when you need to find the solutions to the equation quickly. You should use factoring when the quadratic expression can be factored easily, or when you need to find the factors of the expression.

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

A: The common mistakes when using the quadratic formula include:

  • Incorrectly identifying the coefficients: Make sure to identify the coefficients correctly before plugging them into the quadratic formula.
  • Incorrectly evaluating the solutions: Make sure to evaluate the solutions correctly by plugging them back into the original equation to check if they satisfy the equation.
  • Not simplifying the solutions: Make sure to simplify the solutions correctly to obtain the final answer.

Q: How do I simplify the solutions to the quadratic equation?

A: To simplify the solutions to the quadratic equation, you need to plug the solutions back into the original equation to check if they satisfy the equation. If they do, then you have found the correct solutions.

Q: What are the real-world applications of the quadratic formula?

A: The quadratic formula has numerous real-world applications in various fields such as physics, engineering, and economics. Some examples include:

  • Projectile motion: The quadratic formula can be used to model the trajectory of a projectile under the influence of gravity.
  • Electrical circuits: The quadratic formula can be used to analyze the behavior of electrical circuits.
  • Economics: The quadratic formula can be used to model the behavior of economic systems.

Q: How do I choose between the quadratic formula and other methods for solving quadratic equations?

A: You should choose the method that is most suitable for the problem at hand. If the quadratic expression can be factored easily, then factoring may be the best method. If the quadratic expression cannot be factored easily, then the quadratic formula may be the best method.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. It provides two solutions to the equation, which are the values of x that satisfy the equation. In this article, we have answered some of the most frequently asked questions about the quadratic formula.

Final Answer

The final answer to the quadratic equation 3x^2 - 7x - 1 = 0 is:

x = (7 - √61) / 6 or x = (7 + √61) / 6

References

  • "Quadratic Formula" by Math Open Reference. Math Open Reference, 2022.
  • "Quadratic Equations" by Khan Academy. Khan Academy, 2022.
  • "Solving Quadratic Equations" by Purplemath. Purplemath, 2022.

Glossary

  • Quadratic equation: A polynomial equation of degree two, which means the highest power of the variable is two.
  • Quadratic formula: A powerful tool for solving quadratic equations, which provides two solutions to the equation.
  • Coefficients: The constants in a quadratic equation, which are a, b, and c.
  • Solutions: The values of x that satisfy the quadratic equation.

FAQs

  • What is the quadratic formula? The quadratic formula is a powerful tool for solving quadratic equations, which provides two solutions to the equation.
  • How do I use the quadratic formula? To use the quadratic formula, you need to identify the coefficients of the quadratic equation and plug them into the formula.
  • What are the common mistakes when using the quadratic formula? The common mistakes when using the quadratic formula include incorrectly identifying the coefficients, incorrectly evaluating the solutions, and not simplifying the solutions.