What Is The Positive Solution To The Equation $0 = -x^2 + 2x + 1$?Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$A. $-2 + \sqrt{2}$B. $2 - \sqrt{2}$C. $1 + \sqrt{2}$D. $-1 + \sqrt{2}$

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 focus on solving the quadratic equation $0 = -x^2 + 2x + 1$ using the quadratic formula. We will also explore the concept of positive solutions and how to identify them.

What is a Quadratic Equation?

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. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.

The Quadratic Formula

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

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation. The quadratic formula provides two solutions for the equation, which are given by the plus and minus signs.

Solving the Equation $0 = -x^2 + 2x + 1$

To solve the equation $0 = -x^2 + 2x + 1$, we can use the quadratic formula. We have:

$a = -1$ $b = 2$ $c = 1$

Substituting these values into the quadratic formula, we get:

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

Simplifying the expression under the square root, we get:

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

$x = \frac{-2 \pm \sqrt{8}}{-2}$

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

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

Positive Solution

We are asked to find the positive solution to the equation. Looking at the solutions we obtained, we see that:

$x = 1 + \sqrt{2}$

is the positive solution.

Conclusion

In this article, we solved the quadratic equation $0 = -x^2 + 2x + 1$ using the quadratic formula. We identified the positive solution to the equation, which is $x = 1 + \sqrt{2}$. We hope this article has provided a clear and concise explanation of how to solve quadratic equations and identify positive solutions.

Answer

The positive solution to the equation $0 = -x^2 + 2x + 1$ is:

$1 + \sqrt{2}$

Discussion

• What is the difference between a quadratic equation and a linear equation?
• How do you identify the coefficients of a quadratic equation?
• What is the significance of the quadratic formula in solving quadratic equations?
• Can you think of a real-world application of quadratic equations?

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Quadratic Formula" by Khan Academy
• [3] "Solving Quadratic Equations" by Purplemath

Related Topics

• Solving Linear Equations
• Solving Polynomial Equations
• Quadratic Formula
• Positive Solutions
  Quadratic Equations Q&A: Frequently Asked Questions and Answers ====================================================================

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. In this article, we will address some of the most frequently asked questions about quadratic equations and provide clear and concise answers.

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: How do I identify the coefficients of a quadratic equation?

A: To identify the coefficients of a quadratic equation, you need to look at the equation and identify the values of $a$, $b$, and $c$. For example, in the equation $x^2 + 3x + 2 = 0$, the coefficients are $a = 1$, $b = 3$, and $c = 2$.

Q: What is the quadratic formula?

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

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

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

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

A: To use the quadratic formula to solve a quadratic equation, you need to substitute the values of $a$, $b$, and $c$ into the formula and simplify the expression. For example, to solve the equation $x^2 + 3x + 2 = 0$, you would substitute $a = 1$, $b = 3$, and $c = 2$ into the quadratic formula and simplify the expression.

Q: What is the significance of the quadratic formula in solving quadratic equations?

A: The quadratic formula is a powerful tool for solving quadratic equations because it provides a general solution to the equation, regardless of the values of $a$, $b$, and $c$. This makes it a useful tool for solving quadratic equations in a variety of situations.

Q: Can you think of a real-world application of quadratic equations?

A: Yes, quadratic equations have many real-world applications. For example, they are used in physics to model the motion of objects, in engineering to design buildings and bridges, and in economics to model the behavior of markets.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared variable, while a linear equation does not.

Q: How do I identify the positive solution to a quadratic equation?

A: To identify the positive solution to a quadratic equation, you need to look at the solutions you obtained using the quadratic formula and identify the solution that is greater than zero.

Q: Can you provide an example of a quadratic equation and its solutions?

A: Yes, here is an example of a quadratic equation and its solutions:

Equation: $x^2 + 3x + 2 = 0$

Solutions: $x = -2$ and $x = -1$

In this example, the positive solution is $x = -1$.

Conclusion

In this article, we addressed some of the most frequently asked questions about quadratic equations and provided clear and concise answers. We hope this article has been helpful in clarifying some of the concepts related to quadratic equations.

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Quadratic Formula" by Khan Academy
• [3] "Solving Quadratic Equations" by Purplemath

Related Topics

• Solving Linear Equations
• Solving Polynomial Equations
• Quadratic Formula
• Positive Solutions