Solve The Equation Using The Quadratic Formula: $ 3x^2 - 1 = 5x}$Options A. { \frac{5+\sqrt{13 }{6}, \frac{5-\sqrt{13}}{6}$}$ B. { \frac{-5+\sqrt{37}}{6}, \frac{-5-\sqrt{37}}{6}$}$ C. [$\frac{-5+\sqrt{13}}{6},

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Introduction

The quadratic formula is a powerful tool used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. It is a fundamental concept in algebra and is widely used in various fields such as physics, engineering, and economics. In this article, we will use the quadratic formula to solve the equation 3x2−1=5x3x^2 - 1 = 5x and explore the different options available.

The Quadratic Formula

The quadratic formula is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where aa, bb, and cc are the coefficients of the quadratic equation. In our case, we have:

3x2−1=5x3x^2 - 1 = 5x

We can rewrite this equation in the standard form ax2+bx+c=0ax^2 + bx + c = 0 as:

3x2−5x−1=03x^2 - 5x - 1 = 0

Now, we can identify the coefficients aa, bb, and cc as:

a=3,b=−5,c=−1a = 3, b = -5, c = -1

Applying the Quadratic Formula

We can now apply the quadratic formula to solve for xx:

x=−(−5)±(−5)2−4(3)(−1)2(3)x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(-1)}}{2(3)}

Simplifying the expression, we get:

x=5±25+126x = \frac{5 \pm \sqrt{25 + 12}}{6}

x=5±376x = \frac{5 \pm \sqrt{37}}{6}

Evaluating the Options

We are given three options to choose from:

A. 5+136,5−136\frac{5+\sqrt{13}}{6}, \frac{5-\sqrt{13}}{6}

B. −5+376,−5−376\frac{-5+\sqrt{37}}{6}, \frac{-5-\sqrt{37}}{6}

C. −5+136,−5−136\frac{-5+\sqrt{13}}{6}, \frac{-5-\sqrt{13}}{6}

Let's evaluate each option to see which one matches our solution.

Option A

Option A is given by:

5+136,5−136\frac{5+\sqrt{13}}{6}, \frac{5-\sqrt{13}}{6}

However, our solution is:

x=5±376x = \frac{5 \pm \sqrt{37}}{6}

We can see that the values of xx in option A do not match our solution.

Option B

Option B is given by:

−5+376,−5−376\frac{-5+\sqrt{37}}{6}, \frac{-5-\sqrt{37}}{6}

Comparing this with our solution, we can see that the values of xx in option B match our solution.

Option C

Option C is given by:

−5+136,−5−136\frac{-5+\sqrt{13}}{6}, \frac{-5-\sqrt{13}}{6}

However, our solution is:

x=5±376x = \frac{5 \pm \sqrt{37}}{6}

We can see that the values of xx in option C do not match our solution.

Conclusion

In conclusion, the correct option is B. −5+376,−5−376\frac{-5+\sqrt{37}}{6}, \frac{-5-\sqrt{37}}{6}. This option matches our solution, which was obtained using the quadratic formula.

Final Answer

The final answer is:

B. −5+376,−5−376\frac{-5+\sqrt{37}}{6}, \frac{-5-\sqrt{37}}{6}

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Introduction

The quadratic formula is a powerful tool used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. In our previous article, we used the quadratic formula to solve the equation 3x2−1=5x3x^2 - 1 = 5x and explored the different options available. In this article, we will answer some frequently asked questions about the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. It is given by:

x=−b±b2−4ac2ax = \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 aa, bb, and cc in the quadratic equation. Then, you can plug these values into the quadratic formula and simplify the expression to find the solutions.

Q: What are the steps to solve a quadratic equation using the quadratic formula?

A: The steps to solve a quadratic equation using the quadratic formula are:

  1. Identify the coefficients aa, bb, and cc in the quadratic equation.
  2. Plug these values into the quadratic formula.
  3. Simplify the expression to find the solutions.

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. Factoring involves expressing the quadratic equation as a product of two binomials, while the quadratic formula involves using a formula to find the solutions.

Q: When should I use the quadratic formula?

A: You should use the quadratic formula when the quadratic equation cannot be factored easily or when you need to find the solutions quickly.

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 aa, bb, and cc correctly.
  • Not simplifying the expression correctly.
  • Not checking the solutions for extraneous solutions.

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, which can be expressed in the form a+bia + bi, where aa and bb are real numbers and ii is the imaginary unit.

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

A: To check if the solutions are extraneous, you need to plug the solutions back into the original quadratic equation and check if they satisfy the equation. If they do not satisfy the equation, then they are extraneous solutions.

Conclusion

In conclusion, the quadratic formula is a powerful tool used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. By understanding how to use the quadratic formula and avoiding common mistakes, you can solve quadratic equations quickly and accurately.

Final Tips

  • Make sure to identify the coefficients aa, bb, and cc correctly.
  • Simplify the expression correctly.
  • Check the solutions for extraneous solutions.
  • Use the quadratic formula when the quadratic equation cannot be factored easily or when you need to find the solutions quickly.

By following these tips, you can become proficient in using the quadratic formula and solving quadratic equations with ease.