Solve For X X X . 6 X 2 = 2 X − 1 6x^2 = 2x - 1 6 X 2 = 2 X − 1 A. X = 1 ± I 5 6 X = \frac{1 \pm I \sqrt{5}}{6} X = 6 1 ± I 5 ​ ​ B. X = 1 ± 5 6 X = \frac{1 \pm \sqrt{5}}{6} X = 6 1 ± 5 ​ ​ C. X = 1 ± I 7 6 X = \frac{1 \pm I \sqrt{7}}{6} X = 6 1 ± I 7 ​ ​ D. X = 17 7 6 X = \frac{17 \sqrt{7}}{6} X = 6 17 7 ​ ​

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Introduction

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand the different methods used to solve them. In this article, we will focus on solving the quadratic equation $6x^2 = 2x - 1$ using the quadratic formula. The quadratic formula is a powerful tool that can be used to solve quadratic equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

The Quadratic Formula

The quadratic formula is given by:

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

In this formula, $a$, $b$, and $c$ are the coefficients of the quadratic equation, and $x$ is the variable that we are trying to solve for. The quadratic formula can be used to solve quadratic equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Solving the Quadratic Equation

To solve the quadratic equation $6x^2 = 2x - 1$, we need to rewrite it in the standard form $ax^2 + bx + c = 0$. We can do this by subtracting $2x$ from both sides of the equation and adding $1$ to both sides. This gives us:

$6x^2 - 2x + 1 = 0$

Now that we have the equation in the standard form, we can use the quadratic formula to solve for $x$. We have:

$a = 6$, $b = -2$, and $c = 1$

Plugging these values into the quadratic formula, we get:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(6)(1)}}{2(6)}$

Simplifying this expression, we get:

$x = \frac{2 \pm \sqrt{4 - 24}}{12}$

$x = \frac{2 \pm \sqrt{-20}}{12}$

$x = \frac{2 \pm 2i\sqrt{5}}{12}$

$x = \frac{1 \pm i\sqrt{5}}{6}$

Conclusion

In this article, we solved the quadratic equation $6x^2 = 2x - 1$ using the quadratic formula. We first rewrote the equation in the standard form $ax^2 + bx + c = 0$, and then we used the quadratic formula to solve for $x$. The solution to the equation is $x = \frac{1 \pm i\sqrt{5}}{6}$.

Discussion

The solution to the quadratic equation $6x^2 = 2x - 1$ is $x = \frac{1 \pm i\sqrt{5}}{6}$. This solution is in the form of complex numbers, which is a common occurrence when solving quadratic equations. The complex numbers are in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Comparison with Other Options

Let's compare our solution with the other options given:

• Option A: $x = \frac{1 \pm i\sqrt{5}}{6}$
• Option B: $x = \frac{1 \pm \sqrt{5}}{6}$
• Option C: $x = \frac{1 \pm i\sqrt{7}}{6}$
• Option D: $x = \frac{17\sqrt{7}}{6}$

Our solution matches with option A, which is $x = \frac{1 \pm i\sqrt{5}}{6}$.

Final Answer

The final answer to the quadratic equation $6x^2 = 2x - 1$ is $x = \frac{1 \pm i\sqrt{5}}{6}$.

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Introduction

In our previous article, we solved the quadratic equation $6x^2 = 2x - 1$ using the quadratic formula. We obtained the solution $x = \frac{1 \pm i\sqrt{5}}{6}$. In this article, we will answer some frequently asked questions related to the solution of the quadratic equation.

Q&A

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool that can be used to solve quadratic equations of the form $ax^2 + bx + c = 0$. It is given by:

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

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the quadratic equation. Then, you can plug these values into the quadratic formula to obtain the solution.

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula and factoring are two different methods used to solve 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: Can I use the quadratic formula to solve quadratic equations with complex solutions?

A: Yes, you can use the quadratic formula to solve quadratic equations with complex solutions. The quadratic formula will give you the complex solutions in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: How do I simplify complex solutions?

A: To simplify complex solutions, you can use the following steps:

1. Multiply the numerator and denominator by the conjugate of the denominator.
2. Simplify the expression.
3. Write the solution in the form $a + bi$, where $a$ and $b$ are real numbers.

Q: What is the significance of the imaginary unit $i$?

A: The imaginary unit $i$ is a fundamental concept in mathematics that is used to extend the real numbers to the complex numbers. It is defined as the square root of $-1$, and it is used to represent complex numbers in the form $a + bi$.

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

A: Yes, you can use the quadratic formula to solve quadratic equations with rational solutions. The quadratic formula will give you the rational solutions in the form $a/b$, where $a$ and $b$ are integers.

Q: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, you can use the discriminant $b^2 - 4ac$. If the discriminant is positive, then the equation has two distinct real solutions. If the discriminant is zero, then the equation has one real solution. If the discriminant is negative, then the equation has two complex solutions.

Conclusion

In this article, we answered some frequently asked questions related to the solution of the quadratic equation $6x^2 = 2x - 1$. We covered topics such as the quadratic formula, factoring, complex solutions, and the significance of the imaginary unit $i$. We hope that this article has been helpful in clarifying any doubts that you may have had about the solution of the quadratic equation.

Final Answer

The final answer to the quadratic equation $6x^2 = 2x - 1$ is $x = \frac{1 \pm i\sqrt{5}}{6}$.