Which Of The Following Is A Solution Of $4x^2 = -9x - 4$?A. $\frac{9 \pm \sqrt{17}}{8}$ B. $\frac{-9 \pm \sqrt{145}}{8}$ C. $\frac{9 \pm \sqrt{145}}{8}$ D. $\frac{-9 \pm \sqrt{17}}{8}$

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving the quadratic equation $4x^2 = -9x - 4$ and explore the different solutions that can be obtained.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where a, b, and c are constants.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

Applying the Quadratic Formula

Now, let's apply the quadratic formula to the given equation $4x^2 = -9x - 4$. First, we need to rewrite the equation in the standard form $ax^2 + bx + c = 0$, which gives us:

4x2+9x+4=04x^2 + 9x + 4 = 0

Comparing this with the general form, we can see that a = 4, b = 9, and c = 4.

Plugging in the Values

Now, let's plug in the values of a, b, and c into the quadratic formula:

x=−9±92−4(4)(4)2(4)x = \frac{-9 \pm \sqrt{9^2 - 4(4)(4)}}{2(4)}

Simplifying the Expression

Simplifying the expression under the square root, we get:

x=−9±81−648x = \frac{-9 \pm \sqrt{81 - 64}}{8}

x=−9±178x = \frac{-9 \pm \sqrt{17}}{8}

Evaluating the Solutions

Now, let's evaluate the two possible solutions:

x=−9+178x = \frac{-9 + \sqrt{17}}{8}

x=−9−178x = \frac{-9 - \sqrt{17}}{8}

Conclusion

In conclusion, the solutions to the quadratic equation $4x^2 = -9x - 4$ are:

x=−9±178x = \frac{-9 \pm \sqrt{17}}{8}

These solutions can be obtained using the quadratic formula, which is a powerful tool for solving quadratic equations.

Discussion

The quadratic formula is a fundamental concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics. In this article, we have seen how to apply the quadratic formula to solve a quadratic equation.

Final Thoughts

Solving quadratic equations is a crucial skill for students and professionals alike. By understanding the quadratic formula and applying it to different equations, we can obtain solutions that can be used to model real-world problems.

Recommendations

For those who want to learn more about quadratic equations and the quadratic formula, we recommend the following resources:

  • Khan Academy: Quadratic Equations
  • MIT OpenCourseWare: Quadratic Equations
  • Wolfram MathWorld: Quadratic Formula

Glossary

  • Quadratic Equation: A polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two.
  • Quadratic Formula: A powerful tool for solving quadratic equations, which states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

References

About the Author

The author is a mathematics enthusiast with a passion for teaching and learning. They have a strong background in mathematics and have taught various courses, including algebra and calculus.

Contact Information

For any questions or comments, please feel free to contact the author at author@email.com.

Disclaimer

The information provided in this article is for educational purposes only and should not be considered as professional advice.

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will provide a Q&A guide to help you understand quadratic equations and the quadratic formula.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where a, b, and c are constants.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

x=−b±b2−4ac2ax = \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 in the equation. Then, plug these values into the quadratic formula and simplify the expression.

Q: What is the difference between the two solutions obtained from the quadratic formula?

A: The two solutions obtained from the quadratic formula are:

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

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

The difference between these two solutions is the sign of the square root term. If the square root term is positive, the solutions are real and distinct. If the square root term is negative, the solutions are complex and conjugate.

Q: How do I determine the nature of the solutions?

A: To determine the nature of the solutions, you need to examine the discriminant, which is the expression under the square root term:

b2−4acb^2 - 4ac

If the discriminant is positive, the solutions are real and distinct. If the discriminant is zero, the solutions are real and equal. If the discriminant is negative, the solutions are complex and conjugate.

Q: What are some common mistakes to avoid when applying the quadratic formula?

A: Some common mistakes to avoid when applying the quadratic formula include:

  • Not identifying the values of a, b, and c correctly
  • Not simplifying the expression correctly
  • Not checking the nature of the solutions
  • Not using the correct sign for the square root term

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

A: To use the quadratic formula to solve a quadratic equation, follow these steps:

  1. Identify the values of a, b, and c in the equation.
  2. Plug these values into the quadratic formula.
  3. Simplify the expression.
  4. Check the nature of the solutions.
  5. Write the solutions in the correct form.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have numerous real-world applications, including:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
  • Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By understanding the quadratic formula and applying it to different equations, we can obtain solutions that can be used to model real-world problems.

Final Thoughts

Solving quadratic equations is a challenging task, but with practice and patience, anyone can master it. By following the steps outlined in this article, you can become proficient in solving quadratic equations and apply them to real-world problems.

Recommendations

For those who want to learn more about quadratic equations and the quadratic formula, we recommend the following resources:

  • Khan Academy: Quadratic Equations
  • MIT OpenCourseWare: Quadratic Equations
  • Wolfram MathWorld: Quadratic Formula

Glossary

  • Quadratic Equation: A polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two.
  • Quadratic Formula: A powerful tool for solving quadratic equations, which states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

References

About the Author

The author is a mathematics enthusiast with a passion for teaching and learning. They have a strong background in mathematics and have taught various courses, including algebra and calculus.

Contact Information

For any questions or comments, please feel free to contact the author at author@email.com.

Disclaimer

The information provided in this article is for educational purposes only and should not be considered as professional advice.