Using The Quadratic Formula To Solve 7 X 2 − X = 7 7x^2 - X = 7 7 X 2 − X = 7 , What Are The Values Of X X X ?A. 1 ± 195 I 14 \frac{1 \pm \sqrt{195} I}{14} 14 1 ± 195 ​ I ​ B. 1 ± 197 14 \frac{1 \pm \sqrt{197}}{14} 14 1 ± 197 ​ ​ C. 1 ± 195 14 \frac{1 \pm \sqrt{195}}{14} 14 1 ± 195 ​ ​ D. $\frac{1 \pm

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. The quadratic formula is a powerful tool for solving quadratic equations, and in this article, we will explore how to use it to solve the equation $7x^2 - x = 7$. We will break down the steps involved in using the quadratic formula and provide a clear and concise explanation of the process.

Understanding the Quadratic Formula

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form $ax^2 + bx + c = 0$. The formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

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

Rearranging the Given Equation

To use the quadratic formula, we need to rearrange the given equation $7x^2 - x = 7$ into the standard form $ax^2 + bx + c = 0$. We can do this by subtracting 7 from both sides of the equation, which gives us:

$7x^2 - x - 7 = 0$

Identifying the Coefficients

Now that we have the equation in the standard form, we can identify the coefficients $a$, $b$, and $c$. In this case, $a = 7$, $b = -1$, and $c = -7$.

Substituting the Coefficients into the Quadratic Formula

Now that we have identified the coefficients, we can substitute them into the quadratic formula:

$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(7)(-7)}}{2(7)}$

Simplifying the Expression

We can simplify the expression inside the square root by evaluating the expression $(-1)^2 - 4(7)(-7)$:

$(-1)^2 - 4(7)(-7) = 1 + 196 = 197$

Substituting the Simplified Expression into the Quadratic Formula

Now that we have simplified the expression inside the square root, we can substitute it into the quadratic formula:

$x = \frac{1 \pm \sqrt{197}}{14}$

Conclusion

In this article, we have used the quadratic formula to solve the equation $7x^2 - x = 7$. We have broken down the steps involved in using the quadratic formula and provided a clear and concise explanation of the process. We have also identified the coefficients $a$, $b$, and $c$ and substituted them into the quadratic formula. Finally, we have simplified the expression inside the square root and substituted it into the quadratic formula to obtain the solutions to the equation.

The Final Answer

The final answer to the equation $7x^2 - x = 7$ is:

$x = \frac{1 \pm \sqrt{197}}{14}$

This is the correct answer, and it is option B in the given discussion category.

Why is this the Correct Answer?

This is the correct answer because we have used the quadratic formula to solve the equation $7x^2 - x = 7$. We have broken down the steps involved in using the quadratic formula and provided a clear and concise explanation of the process. We have also identified the coefficients $a$, $b$, and $c$ and substituted them into the quadratic formula. Finally, we have simplified the expression inside the square root and substituted it into the quadratic formula to obtain the solutions to the equation.

What are the Implications of this Answer?

The implications of this answer are that the solutions to the equation $7x^2 - x = 7$ are complex numbers. This is because the expression inside the square root is positive, which means that the solutions are complex numbers. This is an important result in mathematics, and it has implications for many areas of study, including physics, engineering, and computer science.

What are the Applications of this Answer?

The applications of this answer are numerous. For example, in physics, complex numbers are used to describe the behavior of oscillating systems. In engineering, complex numbers are used to design and analyze electrical circuits. In computer science, complex numbers are used to develop algorithms for solving problems in computer graphics and game development.

What are the Limitations of this Answer?

The limitations of this answer are that it only applies to quadratic equations of the form $ax^2 + bx + c = 0$. It does not apply to other types of equations, such as cubic or quartic equations. Additionally, the answer assumes that the coefficients $a$, $b$, and $c$ are real numbers. If the coefficients are complex numbers, then the answer will be different.

What are the Future Directions of this Answer?

The future directions of this answer are numerous. For example, researchers are currently working on developing new algorithms for solving quadratic equations using complex numbers. Additionally, researchers are working on developing new applications for complex numbers in fields such as physics, engineering, and computer science.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. In this article, we have used the quadratic formula to solve the equation $7x^2 - x = 7$. We have broken down the steps involved in using the quadratic formula and provided a clear and concise explanation of the process. We have also identified the coefficients $a$, $b$, and $c$ and substituted them into the quadratic formula. Finally, we have simplified the expression inside the square root and substituted it into the quadratic formula to obtain the solutions to the equation. The final answer is:

$x = \frac{1 \pm \sqrt{197}}{14}$

Introduction

The quadratic formula is a powerful tool for solving quadratic equations, and it can be a bit tricky to understand at first. In this article, we will answer some of the most frequently asked questions about the quadratic formula, including how to use it, what it means, and how it is used in real-world applications.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form $ax^2 + bx + c = 0$. The formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the coefficients $a$, $b$, and $c$ in the quadratic equation. Then, you substitute these values into the quadratic formula and simplify the expression inside the square root. Finally, you take the square root and simplify the expression to obtain the solutions to the equation.

Q: What does the quadratic formula mean?

A: The quadratic formula provides the solutions to a quadratic equation, which means that it gives you the values of $x$ that satisfy the equation. The formula is based on the concept of completing the square, which is a technique for rewriting a quadratic equation in a form that makes it easier to solve.

Q: How is the quadratic formula used in real-world applications?

A: The quadratic formula is used in a wide range of real-world applications, including physics, engineering, and computer science. For example, it is used to describe the motion of objects under the influence of gravity, to design and analyze electrical circuits, and to develop algorithms for solving problems in computer graphics and game development.

Q: What are the limitations of the quadratic formula?

A: The quadratic formula only applies to quadratic equations of the form $ax^2 + bx + c = 0$. It does not apply to other types of equations, such as cubic or quartic equations. Additionally, the formula assumes that the coefficients $a$, $b$, and $c$ are real numbers. If the coefficients are complex numbers, then the formula will not work.

Q: Can I use the quadratic formula to solve equations with complex coefficients?

A: No, the quadratic formula is not designed to handle equations with complex coefficients. If you have an equation with complex coefficients, you will need to use a different method to solve it.

Q: How do I know if the quadratic formula will work for a given equation?

A: To determine if the quadratic formula will work for a given equation, you need to check if the equation is in the form $ax^2 + bx + c = 0$. If it is, then you can use the quadratic formula to solve the equation. If it is not, then you will need to use a different method to solve the equation.

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 identifying the coefficients $a$, $b$, and $c$ correctly
• Not simplifying the expression inside the square root correctly
• Not taking the square root correctly
• Not simplifying the expression to obtain the solutions to the equation correctly

Q: How do I check my work when using the quadratic formula?

A: To check your work when using the quadratic formula, you can plug the solutions back into the original equation to see if they are true. You can also use a calculator or computer program to check your work.

Q: What are some real-world examples of the quadratic formula in action?

A: Some real-world examples of the quadratic formula in action include:

• Describing the motion of objects under the influence of gravity
• Designing and analyzing electrical circuits
• Developing algorithms for solving problems in computer graphics and game development
• Modeling population growth and decline
• Solving problems in physics, engineering, and computer science

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations, and it has many real-world applications. By understanding how to use the quadratic formula and what it means, you can solve a wide range of problems in physics, engineering, and computer science. Remember to check your work and avoid common mistakes when using the quadratic formula.