Solve:${ 2 - X - X^2 \leq 0 }$

Mar 13, 2025 by ADMIN 32 views

$Solve: $2 - x - x^2 \leq 0$$

Introduction

In this article, we will delve into the world of quadratic inequalities and explore the solution to the given inequality: $2 - x - x^2 \leq 0$ . Quadratic inequalities are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. The solution to this inequality will involve factoring, graphing, and using the properties of quadratic functions.

Understanding the Inequality

The given inequality is a quadratic inequality in the form of $ax^2 + bx + c \leq 0$ , where $a = -1$ , $b = -1$ , and $c = 2$ . To solve this inequality, we need to find the values of $x$ that satisfy the given condition. We can start by rewriting the inequality as $-x^2 - x + 2 \leq 0$ .

Factoring the Quadratic Expression

To factor the quadratic expression $-x^2 - x + 2$ , we need to find two numbers whose product is $-2$ and whose sum is $-1$ . These numbers are $-2$ and $1$ , so we can write the quadratic expression as $-(x^2 + x - 2)$ . We can then factor the quadratic expression as $-(x + 2)(x - 1)$ .

Graphing the Quadratic Function

To graph the quadratic function $y = -(x + 2)(x - 1)$ , we need to find the x-intercepts and the vertex of the parabola. The x-intercepts are the values of $x$ that make the function equal to zero. We can find the x-intercepts by setting the function equal to zero and solving for $x$ . This gives us the x-intercepts $x = -2$ and $x = 1$ . The vertex of the parabola is the point where the function changes from decreasing to increasing or vice versa. We can find the vertex by using the formula $x = -\frac{b}{2a}$ , where $a$ and $b$ are the coefficients of the quadratic function. This gives us the vertex $x = -\frac{1}{2}$ .

Solving the Inequality

To solve the inequality $-(x + 2)(x - 1) \leq 0$ , we need to find the values of $x$ that satisfy the given condition. We can start by finding the intervals where the function is positive or negative. We can do this by testing a value of $x$ in each interval and determining whether the function is positive or negative. We can then use this information to determine the solution to the inequality.

Solution to the Inequality

The solution to the inequality $-(x + 2)(x - 1) \leq 0$ is the interval $[-2, 1]$ . This means that the values of $x$ that satisfy the inequality are $x \in [-2, 1]$ .

Conclusion

In this article, we have solved the quadratic inequality $2 - x - x^2 \leq 0$ . We have factored the quadratic expression, graphed the quadratic function, and used the properties of quadratic functions to find the solution to the inequality. The solution to the inequality is the interval $[-2, 1]$ , which means that the values of $x$ that satisfy the inequality are $x \in [-2, 1]$ .

Final Answer

The final answer to the inequality $2 - x - x^2 \leq 0$ is $x \in [-2, 1]$ .

Additional Resources

For more information on quadratic inequalities and their solutions, please refer to the following resources:

Frequently Asked Questions

Q: What is the solution to the inequality $2 - x - x^2 \leq 0$ ? A: The solution to the inequality is $x \in [-2, 1]$ .
Q: How do I factor the quadratic expression $-x^2 - x + 2$ ? A: You can factor the quadratic expression as $-(x + 2)(x - 1)$ .
Q: How do I graph the quadratic function $y = -(x + 2)(x - 1)$ ? A: You can graph the quadratic function by finding the x-intercepts and the vertex of the parabola.

Step-by-Step Solution

Factor the quadratic expression $-x^2 - x + 2$ as $-(x + 2)(x - 1)$ .
Graph the quadratic function $y = -(x + 2)(x - 1)$ by finding the x-intercepts and the vertex of the parabola.
Solve the inequality $-(x + 2)(x - 1) \leq 0$ by finding the intervals where the function is positive or negative.
Determine the solution to the inequality by using the information from step 3.

Key Takeaways

The solution to the inequality $2 - x - x^2 \leq 0$ is $x \in [-2, 1]$ .
The quadratic expression $-x^2 - x + 2$ can be factored as $-(x + 2)(x - 1)$ .
The quadratic function $y = -(x + 2)(x - 1)$ can be graphed by finding the x-intercepts and the vertex of the parabola.

Summary

Introduction

In our previous article, we solved the quadratic inequality $2 - x - x^2 \leq 0$ . We factored the quadratic expression, graphed the quadratic function, and used the properties of quadratic functions to find the solution to the inequality. In this article, we will answer some frequently asked questions about quadratic inequalities and provide additional resources for further learning.

Q&A

Q: What is a quadratic inequality?

A: A quadratic inequality is an inequality that involves a quadratic expression, which is a polynomial of degree two. The general form of a quadratic inequality is $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to find the values of $x$ that satisfy the given condition. You can do this by factoring the quadratic expression, graphing the quadratic function, and using the properties of quadratic functions.

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

A: A quadratic equation is an equation that involves a quadratic expression, which is a polynomial of degree two. A quadratic inequality, on the other hand, is an inequality that involves a quadratic expression. The main difference between the two is that a quadratic equation has a specific solution, while a quadratic inequality has a range of solutions.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to find the x-intercepts and the vertex of the parabola. You can do this by using the formula $x = -\frac{b}{2a}$ , where $a$ and $b$ are the coefficients of the quadratic function.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the function changes from decreasing to increasing or vice versa. It is the lowest or highest point on the graph of the function.

Q: How do I find the x-intercepts of a quadratic function?

A: To find the x-intercepts of a quadratic function, you need to set the function equal to zero and solve for $x$ . This will give you the x-intercepts of the parabola.

Q: What is the significance of the x-intercepts of a quadratic function?

A: The x-intercepts of a quadratic function are the points where the function intersects the x-axis. They are also the solutions to the quadratic equation.

Q: How do I determine the solution to a quadratic inequality?

A: To determine the solution to a quadratic inequality, you need to find the intervals where the function is positive or negative. You can do this by testing a value of $x$ in each interval and determining whether the function is positive or negative.

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

A: A linear inequality is an inequality that involves a linear expression, which is a polynomial of degree one. A quadratic inequality, on the other hand, is an inequality that involves a quadratic expression. The main difference between the two is that a linear inequality has a specific solution, while a quadratic inequality has a range of solutions.

Q: How do I solve a system of quadratic inequalities?

A: To solve a system of quadratic inequalities, you need to find the values of $x$ that satisfy all the given conditions. You can do this by solving each inequality separately and then finding the intersection of the solutions.