Which Two Values Of $x$ Are Roots Of The Polynomial Below?$x^2 - 11x + 17$A. \$x = 2.5$[/tex\] B. $x = \frac{11 - \sqrt{53}}{2}$ C. $x = \frac{11 - \sqrt{-109}}{4}$ D. \$x = \frac{11

Introduction

In mathematics, 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. In this article, we will focus on solving quadratic equations of the form $x^2 - 11x + 17 = 0$ and finding the roots of the polynomial.

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:

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

In our case, we have $a = 1$, $b = -11$, and $c = 17$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(1)(17)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{11 \pm \sqrt{121 - 68}}{2}$$

$$x = \frac{11 \pm \sqrt{53}}{2}$$

Analyzing the Solutions

We have two possible solutions for $x$:

$$x = \frac{11 + \sqrt{53}}{2}$$

$$x = \frac{11 - \sqrt{53}}{2}$$

Let's analyze these solutions to see which one is correct.

Option A: $x = 2.5$

Option A states that $x = 2.5$ is a root of the polynomial. However, we can see that this solution does not satisfy the quadratic equation. Plugging $x = 2.5$ into the equation, we get:

$$(2.5)^2 - 11(2.5) + 17 = 6.25 - 27.5 + 17 = -4.25$$

Since the result is not equal to zero, $x = 2.5$ is not a root of the polynomial.

Option B: $x = \frac{11 - \sqrt{53}}{2}$

Option B states that $x = \frac{11 - \sqrt{53}}{2}$ is a root of the polynomial. Let's plug this solution into the equation to see if it satisfies the equation:

$$(\frac{11 - \sqrt{53}}{2})^2 - 11(\frac{11 - \sqrt{53}}{2}) + 17 = \frac{121 - 22\sqrt{53} + 53}{4} - \frac{121 - 11\sqrt{53}}{2} + 17$$

Simplifying the expression, we get:

$$\frac{174 - 22\sqrt{53}}{4} - \frac{121 - 11\sqrt{53}}{2} + 17 = \frac{174 - 22\sqrt{53} - 242 + 22\sqrt{53}}{4} + \frac{34}{2}$$

$$= \frac{-68}{4} + 17 = -17 + 17 = 0$$

Since the result is equal to zero, $x = \frac{11 - \sqrt{53}}{2}$ is indeed a root of the polynomial.

Option C: $x = \frac{11 - \sqrt{-109}}{4}$

Option C states that $x = \frac{11 - \sqrt{-109}}{4}$ is a root of the polynomial. However, we can see that this solution is not valid because the square root of a negative number is not a real number. Therefore, this option is not a root of the polynomial.

Conclusion

In conclusion, we have found that the two values of $x$ that are roots of the polynomial $x^2 - 11x + 17 = 0$ are:

$$x = \frac{11 - \sqrt{53}}{2}$$

$$x = \frac{11 + \sqrt{53}}{2}$$

These solutions satisfy the quadratic equation and are therefore the correct roots of the polynomial.

References

• [1] "Quadratic Formula" by Math Is Fun. Retrieved from https://www.mathsisfun.com/algebra/quadratic-formula.html
• [2] "Solving Quadratic Equations" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f-quadratic-equations/x2f-quadratic-equations-article

Note

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Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding them is crucial for solving a wide range of problems in various fields. In this article, we will provide a comprehensive Q&A guide to help you understand quadratic equations and how to solve them.

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, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and completing the square. The most common method is to use the quadratic formula, which is:

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

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:

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

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. For example, if you have the equation $x^2 + 5x + 6 = 0$, you would plug in $a = 1$, $b = 5$, and $c = 6$ into the formula.

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

A: The two solutions of the quadratic formula are given by:

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

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

The difference between the 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 if a quadratic equation has real or complex solutions?

A: To determine if a quadratic equation has real or complex solutions, you need to check the discriminant, which is the expression under the square root in the quadratic formula. 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 is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula, which is:

$$b^2 - 4ac$$

Q: How do I use the discriminant to determine the nature of the solutions?

A: To use the discriminant to determine the nature of the solutions, you need to check the sign of the discriminant. 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 the applications of quadratic equations?

A: Quadratic equations have numerous applications in various fields, including physics, engineering, economics, and computer science. Some examples of applications include:

• Modeling the motion of objects under the influence of gravity
• Determining the maximum or minimum value of a function
• Finding the roots of a polynomial equation
• Solving systems of linear equations

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and understanding them is crucial for solving a wide range of problems in various fields. This Q&A guide has provided a comprehensive overview of quadratic equations, including how to solve them using the quadratic formula and how to determine the nature of the solutions using the discriminant. We hope that this guide has been helpful in understanding quadratic equations and their applications.