Given The Equation X 2 − 2 X − 9 = 0 X^2 - 2x - 9 = 0 X 2 − 2 X − 9 = 0 , One Solution Can Be Written As 1 + K 1 + \sqrt{k} 1 + K ​ , Where K K K Is A Constant. What Is The Value Of K K K ?A. 8 B. 10 C. 20 D. 40

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 a specific quadratic equation, $x^2 - 2x - 9 = 0$, and finding the value of a constant $k$ that is related to one of its solutions.

Understanding the Quadratic Equation

The given quadratic equation is $x^2 - 2x - 9 = 0$. This equation is in the standard form of a quadratic equation, $ax^2 + bx + c = 0$, where $a = 1$, $b = -2$, and $c = -9$. To solve this equation, we can use various methods, including factoring, completing the square, or using the quadratic formula.

Using the Quadratic Formula

One of the most common methods for solving quadratic equations is the quadratic formula. The quadratic formula is given by:

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

In this case, $a = 1$, $b = -2$, and $c = -9$. Plugging these values into the quadratic formula, we get:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-9)}}{2(1)}$

Simplifying the expression, we get:

$x = \frac{2 \pm \sqrt{4 + 36}}{2}$

$x = \frac{2 \pm \sqrt{40}}{2}$

$x = \frac{2 \pm 2\sqrt{10}}{2}$

$x = 1 \pm \sqrt{10}$

Finding the Value of k

We are given that one solution can be written as $1 + \sqrt{k}$, where $k$ is a constant. Comparing this with the solution we found using the quadratic formula, $x = 1 \pm \sqrt{10}$, we can see that $k = 10$.

Conclusion

In this article, we solved the quadratic equation $x^2 - 2x - 9 = 0$ using the quadratic formula and found that one of its solutions is $1 + \sqrt{10}$. We then compared this solution with the given solution $1 + \sqrt{k}$ and found that $k = 10$. Therefore, the value of $k$ is $\boxed{10}$.

Discussion

The value of $k$ is a constant that is related to one of the solutions of the quadratic equation. In this case, we found that $k = 10$. This value can be used to find other solutions of the equation or to verify the solution we found.

Related Topics

• Quadratic Equations: Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike.
• Quadratic Formula: The quadratic formula is a method for solving quadratic equations, and it is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Solving Quadratic Equations: Solving quadratic equations is a crucial skill for students and professionals alike, and it involves using various methods, including factoring, completing the square, or using the quadratic formula.

Conclusion

In conclusion, solving quadratic equations is a crucial skill for students and professionals alike, and it involves using various methods, including factoring, completing the square, or using the quadratic formula. In this article, we solved the quadratic equation $x^2 - 2x - 9 = 0$ using the quadratic formula and found that one of its solutions is $1 + \sqrt{10}$. We then compared this solution with the given solution $1 + \sqrt{k}$ and found that $k = 10$. Therefore, the value of $k$ is $\boxed{10}$.

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 comprehensive guide to quadratic equations, including a detailed solution to the equation $x^2 - 2x - 9 = 0$ and a Q&A section to address common questions and concerns.

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, and $x$ is the variable.

Solving Quadratic Equations

There are several methods for solving quadratic equations, including:

• Factoring: This method involves expressing the quadratic equation as a product of two binomials.
• Completing the Square: This method involves manipulating the quadratic equation to express it in the form $(x + p)^2 = q$.
• Quadratic Formula: This method involves using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions of the quadratic equation.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Solving the Equation $x^2 - 2x - 9 = 0$

In this article, we solved the quadratic equation $x^2 - 2x - 9 = 0$ using the quadratic formula. We found that one of its solutions is $1 + \sqrt{10}$.

Q&A

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.

Q: How do I solve a quadratic equation?

A: There are several methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a method for solving quadratic equations, and it is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula and simplify.

Q: What is the difference between factoring and completing the square?

A: Factoring involves expressing the quadratic equation as a product of two binomials, while completing the square involves manipulating the quadratic equation to express it in the form $(x + p)^2 = q$.

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.

Q: What is the value of $k$ in the equation $1 + \sqrt{k}$?

A: In the equation $1 + \sqrt{k}$, the value of $k$ is 10.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we provided a comprehensive guide to quadratic equations, including a detailed solution to the equation $x^2 - 2x - 9 = 0$ and a Q&A section to address common questions and concerns. We hope this article has been helpful in understanding quadratic equations and how to solve them.