What Are The Solutions Of The Quadratic Equation Below? 2 X 2 − 2 X − 9 = 0 2x^2 - 2x - 9 = 0 2 X 2 − 2 X − 9 = 0 A. − 1 ± 19 2 \frac{-1 \pm \sqrt{19}}{2} 2 − 1 ± 19 ​ ​ B. 3 ± 19 2 \frac{3 \pm \sqrt{19}}{2} 2 3 ± 19 ​ ​ C. 1 ± 19 2 \frac{1 \pm \sqrt{19}}{2} 2 1 ± 19 ​ ​ D. 1 ± 3 19 2 \frac{1 \pm 3\sqrt{19}}{2} 2 1 ± 3 19 ​ ​

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 $2x^2 - 2x - 9 = 0$. We will use the quadratic formula to find the solutions and compare them with the given options.

Understanding Quadratic Equations

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. In our case, the equation is $2x^2 - 2x - 9 = 0$, where $a = 2$, $b = -2$, and $c = -9$.

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 will give us two solutions for the quadratic equation, which are the values of $x$ that satisfy the equation.

Applying the Quadratic Formula

Now, let's apply the quadratic formula to our equation $2x^2 - 2x - 9 = 0$. We have $a = 2$, $b = -2$, and $c = -9$. Plugging these values into the quadratic formula, we get:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(2)(-9)}}{2(2)}$

Simplifying the expression, we get:

$x = \frac{2 \pm \sqrt{4 + 72}}{4}$

$x = \frac{2 \pm \sqrt{76}}{4}$

$x = \frac{2 \pm 2\sqrt{19}}{4}$

$x = \frac{1 \pm \sqrt{19}}{2}$

Comparing with the Given Options

Now, let's compare our solutions with the given options:

A. $\frac{-1 \pm \sqrt{19}}{2}$ B. $\frac{3 \pm \sqrt{19}}{2}$ C. $\frac{1 \pm \sqrt{19}}{2}$ D. $\frac{1 \pm 3\sqrt{19}}{2}$

Our solutions match with option C: $\frac{1 \pm \sqrt{19}}{2}$.

Conclusion

In this article, we solved the quadratic equation $2x^2 - 2x - 9 = 0$ using the quadratic formula. We obtained two solutions, which matched with option C: $\frac{1 \pm \sqrt{19}}{2}$. This demonstrates the power of the quadratic formula in solving quadratic equations.

Final Thoughts

Solving quadratic equations is an essential skill in mathematics, and the quadratic formula is a powerful tool for doing so. By applying the quadratic formula, we can find the solutions to quadratic equations and compare them with given options. This article has demonstrated the step-by-step process of solving a quadratic equation using the quadratic formula.

Additional Resources

For more information on quadratic equations and the quadratic formula, please refer to the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Formula
• Wolfram Alpha: Quadratic Formula

FAQs

Q: What is the quadratic formula? A: The quadratic formula is a powerful tool for solving quadratic equations. 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 plug in the values of $a$, $b$, and $c$ into the formula and simplify the expression.

Q: What are the solutions to the quadratic equation $2x^2 - 2x - 9 = 0$? A: The solutions to the quadratic equation $2x^2 - 2x - 9 = 0$ are $\frac{1 \pm \sqrt{19}}{2}$.

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 answer some frequently asked questions about 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 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 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 plug in the values of $a$, $b$, and $c$ into the formula and simplify the expression. Here's a step-by-step guide:

1. Identify the values of $a$, $b$, and $c$ in the quadratic equation.
2. Plug these values into the quadratic formula.
3. Simplify the expression by combining like terms.
4. Solve for $x$ by isolating the variable.

Q: What are the solutions to the quadratic equation $2x^2 - 2x - 9 = 0$?

A: The solutions to the quadratic equation $2x^2 - 2x - 9 = 0$ are $\frac{1 \pm \sqrt{19}}{2}$.

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

A: Yes, the quadratic formula can be used to solve any quadratic equation in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: What if the quadratic equation has no real solutions?

A: If the quadratic equation has no real solutions, it means that the discriminant ($b^2 - 4ac$) is negative. In this case, the quadratic formula will give you complex solutions.

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

A: No, the quadratic formula is only applicable to quadratic equations with real coefficients. If the coefficients are complex, you will need to use a different method to solve the equation.

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, the equation has two distinct real solutions.
• Zero, the equation has one real solution.
• Negative, the equation has no real solutions.

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.

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

A: Yes, the quadratic formula can be used to solve quadratic equations with irrational coefficients.

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 simplifying the expression correctly.
• Not isolating the variable correctly.
• Not checking the discriminant to determine the number of solutions.

Conclusion

In this article, we have answered some frequently asked questions about quadratic equations and the quadratic formula. We hope that this article has provided you with a better understanding of quadratic equations and how to solve them using the quadratic formula.

Additional Resources

For more information on quadratic equations and the quadratic formula, please refer to the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Formula
• Wolfram Alpha: Quadratic Formula

FAQs

Q: What is the quadratic formula? A: The quadratic formula is a powerful tool for solving quadratic equations. 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 plug in the values of $a$, $b$, and $c$ into the formula and simplify the expression.

Q: What are the solutions to the quadratic equation $2x^2 - 2x - 9 = 0$? A: The solutions to the quadratic equation $2x^2 - 2x - 9 = 0$ are $\frac{1 \pm \sqrt{19}}{2}$.