Solve $3x - 1 = \frac{1}{x}$ Using The Quadratic Formula.

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Introduction

The quadratic formula is a powerful tool for solving quadratic equations. However, it can also be used to solve equations that are not quadratic in the classical sense. In this article, we will explore how to use the quadratic formula to solve the equation $3x - 1 = \frac{1}{x}$.

Understanding the Equation

The given equation is $3x - 1 = \frac{1}{x}$. To solve this equation using the quadratic formula, we need to first manipulate it into a quadratic form. We can do this by multiplying both sides of the equation by $x$, which gives us $3x^2 - x = 1$.

Rearranging the Equation

Next, we need to rearrange the equation to get it into the standard quadratic form $ax^2 + bx + c = 0$. We can do this by subtracting $1$ from both sides of the equation, which gives us $3x^2 - x - 1 = 0$.

Applying the Quadratic Formula

Now that we have the equation in the standard quadratic form, we can apply the quadratic formula to solve for $x$. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

In this case, we have $a = 3$, $b = -1$, and $c = -1$. Plugging these values into the quadratic formula, we get:

x=−(−1)±(−1)2−4(3)(−1)2(3)x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(-1)}}{2(3)}

Simplifying the Expression

Simplifying the expression inside the square root, we get:

x=1±1+126x = \frac{1 \pm \sqrt{1 + 12}}{6}

x=1±136x = \frac{1 \pm \sqrt{13}}{6}

Finding the Solutions

Now that we have the simplified expression, we can find the two solutions for $x$. We can do this by plugging in the $\pm$ sign and solving for $x$.

The two solutions are:

x=1+136x = \frac{1 + \sqrt{13}}{6}

x=1−136x = \frac{1 - \sqrt{13}}{6}

Conclusion

In this article, we have shown how to use the quadratic formula to solve the equation $3x - 1 = \frac{1}{x}$. We first manipulated the equation into a quadratic form, then applied the quadratic formula to solve for $x$. The two solutions to the equation are $x = \frac{1 + \sqrt{13}}{6}$ and $x = \frac{1 - \sqrt{13}}{6}$.

Final Thoughts

The quadratic formula is a powerful tool for solving quadratic equations. However, it can also be used to solve equations that are not quadratic in the classical sense. By manipulating the equation into a quadratic form and applying the quadratic formula, we can solve equations that would otherwise be difficult or impossible to solve.

References

Additional Resources

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Frequently Asked Questions

Q: What is the quadratic formula and how is it used to solve equations?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I use the quadratic formula to solve the equation $3x - 1 = \frac{1}{x}$?

A: To use the quadratic formula to solve the equation $3x - 1 = \frac{1}{x}$, you need to first manipulate the equation into a quadratic form. You can do this by multiplying both sides of the equation by $x$, which gives you $3x^2 - x = 1$. Then, you need to rearrange the equation to get it into the standard quadratic form $ax^2 + bx + c = 0$. You can do this by subtracting $1$ from both sides of the equation, which gives you $3x^2 - x - 1 = 0$. Finally, you can apply the quadratic formula to solve for $x$.

Q: What are the two solutions to the equation $3x - 1 = \frac{1}{x}$?

A: The two solutions to the equation $3x - 1 = \frac{1}{x}$ are $x = \frac{1 + \sqrt{13}}{6}$ and $x = \frac{1 - \sqrt{13}}{6}$.

Q: How do I know which solution to use?

A: You can use either solution to the equation $3x - 1 = \frac{1}{x}$. However, you need to check that the solution you choose is valid by plugging it back into the original equation.

Q: Can I use the quadratic formula to solve any equation?

A: No, the quadratic formula can only be used to solve quadratic equations of the form $ax^2 + bx + c = 0$. If you have an equation that is not quadratic, you may need to use a different method to solve it.

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 rearranging the equation into the standard quadratic form $ax^2 + bx + c = 0$.
  • Not plugging in the correct values for $a$, $b$, and $c$ into the quadratic formula.
  • Not checking that the solution you choose is valid by plugging it back into the original equation.

Additional Resources

Conclusion

In this article, we have answered some frequently asked questions about solving the equation $3x - 1 = \frac{1}{x}$ using the quadratic formula. We have also provided some additional resources for further learning and practice.