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

Mar 1, 2025 by ADMIN 194 views

**Solving Quadratic Equations Using the Quadratic Formula**

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.

The Quadratic Formula

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

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

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

Solving the Given Quadratic Equation

We are given the quadratic equation $7x^2 - x = 7$ . To solve this equation using the quadratic formula, we need to rewrite it in the standard form $ax^2 + bx + c = 0$ . We can do this by subtracting $7$ from both sides of the equation:

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

Now, we can identify the values of $a$ , $b$ , and $c$ :

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

Applying the Quadratic Formula

We can now apply the quadratic formula to solve the equation:

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

Simplifying the expression, we get:

$x = \frac{1 \pm \sqrt{1 + 196}}{14}$

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

Evaluating the Solutions

We have obtained two solutions for the quadratic equation:

$x = \frac{1 + \sqrt{197}}{14}$ and $x = \frac{1 - \sqrt{197}}{14}$

These are the values of $x$ that satisfy the equation.

Conclusion

In this article, we have used the quadratic formula to solve the quadratic equation $7x^2 - x = 7$ . We have identified the values of $a$ , $b$ , and $c$ , and applied the quadratic formula to obtain the solutions. The solutions are $x = \frac{1 + \sqrt{197}}{14}$ and $x = \frac{1 - \sqrt{197}}{14}$ .

Comparison with the Given Options

We can compare our solutions with the given options:

A. $\frac{1 \pm \sqrt{195} i}{14}$

B. $\frac{1 \pm \sqrt{197}}{14}$

C. $\frac{1 \pm \sqrt{195}}{14}$

D. $\frac{1 \pm \sqrt{195} i}{14}$

Our solutions match with option B.

Final Answer

The final answer is $\boxed{B}$ .

Additional Information

The quadratic formula is a powerful tool for solving quadratic equations. It provides two solutions for the equation, which are the values of $x$ that satisfy the equation. The quadratic formula is given by:

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

This formula can be used to solve quadratic equations in the form $ax^2 + bx + c = 0$ . The values of $a$ , $b$ , and $c$ can be identified from the equation, and the quadratic formula can be applied to obtain the solutions.

Real-World Applications

Quadratic equations have numerous applications in various fields such as physics, engineering, and economics. For example, the trajectory of a projectile under the influence of gravity can be modeled using a quadratic equation. The quadratic formula can be used to solve the equation and obtain the values of the variables.

Tips and Tricks

When using the quadratic formula, it is essential to identify the values of $a$ , $b$ , and $c$ correctly. This can be done by rewriting the equation in the standard form $ax^2 + bx + c = 0$ . The quadratic formula can then be applied to obtain the solutions.

Common Mistakes

One common mistake when using the quadratic formula is to forget to simplify the expression under the square root. This can lead to incorrect solutions. It is essential to simplify the expression correctly to obtain the correct solutions.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. It provides two solutions for the equation, which are the values of $x$ that satisfy the equation. The quadratic formula can be used to solve quadratic equations in the form $ax^2 + bx + c = 0$ . The values of $a$ , $b$ , and $c$ can be identified from the equation, and the quadratic formula can be applied to obtain the solutions.

Introduction

The quadratic formula is a powerful tool for solving quadratic equations. However, it can be a bit tricky to understand and apply. 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 mathematical formula that provides two solutions for a quadratic equation in the form $ax^2 + bx + c = 0$ . It is given by:

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

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, you need to identify the values of $a$ , $b$ , and $c$ from the quadratic equation. Then, you can plug these values into the quadratic formula and simplify the expression to obtain the solutions.

Q: What is the difference between the two solutions?

A: The two solutions obtained from the quadratic formula are the values of $x$ that satisfy the equation. They are usually denoted as $x_1$ and $x_2$ . The difference between the two solutions is the value of the expression under the square root, which is $b^2 - 4ac$ .

Q: Can I use the quadratic formula to solve all quadratic equations?

A: Yes, the quadratic formula can be used to solve all quadratic equations in the form $ax^2 + bx + c = 0$ . However, it is essential to identify the values of $a$ , $b$ , and $c$ correctly and to simplify the expression under the square root.

Q: What happens if the expression under the square root is negative?

A: If the expression under the square root is negative, then the quadratic equation has no real solutions. In this case, the solutions will be complex numbers.

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

A: Yes, the quadratic formula can be used to solve quadratic equations with complex coefficients. However, the solutions will be complex numbers.

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

A: To simplify the expression under the square root, you need to factor the expression and simplify it as much as possible. You can also use the quadratic formula to simplify the expression.

Q: What is the significance of the quadratic formula in real-world applications?

A: The quadratic formula has numerous applications in various fields such as physics, engineering, and economics. For example, the trajectory of a projectile under the influence of gravity can be modeled using a quadratic equation. The quadratic formula can be used to solve the equation and obtain the values of the variables.

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

A: Yes, the quadratic formula can be used to solve quadratic equations with rational coefficients. However, the solutions will be rational numbers.

Q: How do I check if the solutions are correct?

A: To check if the solutions are correct, you need to plug the solutions back into the original equation and simplify the expression. If the expression is equal to zero, then the solutions are correct.

Conclusion

Additional Resources

For more information on the quadratic formula, you can refer to the following resources:

Final Answer

The final answer is $\boxed{B}$ .