Solve The Following Inequality. X 2 − 4 X + 8 ≤ 0 X^2 - 4x + 8 \leq 0 X 2 − 4 X + 8 ≤ 0 Select The Correct Choice Below, And If Necessary, Fill In The Answer Box To Complete Your Choice:A. The Solution Is □ \square □ . (Type Your Answer In Interval Notation.)B. There Is

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Introduction

In this article, we will focus on solving the given inequality $x^2 - 4x + 8 \leq 0$. This involves finding the values of $x$ that satisfy the given inequality. We will use various mathematical techniques to solve this inequality and provide the solution in interval notation.

Understanding the Inequality

The given inequality is a quadratic inequality in the form of $ax^2 + bx + c \leq 0$. In this case, $a = 1$, $b = -4$, and $c = 8$. To solve this inequality, we need to find the values of $x$ that make the quadratic expression $x^2 - 4x + 8$ less than or equal to zero.

Factoring the Quadratic Expression

One way to solve the inequality is to factor the quadratic expression $x^2 - 4x + 8$. However, this expression does not factor easily. Therefore, we will use the quadratic formula to find the roots of the quadratic equation $x^2 - 4x + 8 = 0$.

Using the Quadratic Formula

The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, $a = 1$, $b = -4$, and $c = 8$. Plugging these values into the quadratic formula, we get:

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(8)}}{2(1)}$

$x = \frac{4 \pm \sqrt{16 - 32}}{2}$

$x = \frac{4 \pm \sqrt{-16}}{2}$

$x = \frac{4 \pm 4i}{2}$

$x = 2 \pm 2i$

Finding the Roots of the Quadratic Equation

The roots of the quadratic equation $x^2 - 4x + 8 = 0$ are given by $x = 2 \pm 2i$. These roots are complex numbers, which means that the quadratic expression $x^2 - 4x + 8$ does not have any real roots.

Understanding the Sign of the Quadratic Expression

Since the quadratic expression $x^2 - 4x + 8$ does not have any real roots, we need to understand the sign of the quadratic expression for different values of $x$. We can do this by using a sign chart or by graphing the quadratic function.

Creating a Sign Chart

To create a sign chart, we need to find the intervals where the quadratic expression $x^2 - 4x + 8$ is positive, negative, or zero. We can do this by finding the intervals where the quadratic expression is greater than, less than, or equal to zero.

Graphing the Quadratic Function

We can graph the quadratic function $y = x^2 - 4x + 8$ to visualize the sign of the quadratic expression for different values of $x$. The graph of the quadratic function will help us understand the intervals where the quadratic expression is positive, negative, or zero.

Understanding the Sign of the Quadratic Expression

From the sign chart or the graph of the quadratic function, we can see that the quadratic expression $x^2 - 4x + 8$ is negative for $x < 2 - 2i$ and $x > 2 + 2i$. This means that the inequality $x^2 - 4x + 8 \leq 0$ is satisfied for $x < 2 - 2i$ and $x > 2 + 2i$.

Writing the Solution in Interval Notation

The solution to the inequality $x^2 - 4x + 8 \leq 0$ is given by the intervals where the quadratic expression is negative. Therefore, the solution is:

$(-\infty, 2 - 2i) \cup (2 + 2i, \infty)$

Conclusion

In this article, we solved the inequality $x^2 - 4x + 8 \leq 0$ by using various mathematical techniques. We found the roots of the quadratic equation $x^2 - 4x + 8 = 0$ using the quadratic formula and understood the sign of the quadratic expression for different values of $x$. We then wrote the solution to the inequality in interval notation.

Final Answer

The final answer is: $\boxed{(-\infty, 2 - 2i) \cup (2 + 2i, \infty)}$

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Introduction

In our previous article, we solved the inequality $x^2 - 4x + 8 \leq 0$ by using various mathematical techniques. We found the roots of the quadratic equation $x^2 - 4x + 8 = 0$ using the quadratic formula and understood the sign of the quadratic expression for different values of $x$. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q: What is the solution to the inequality $x^2 - 4x + 8 \leq 0$?

A: The solution to the inequality $x^2 - 4x + 8 \leq 0$ is given by the intervals where the quadratic expression is negative. Therefore, the solution is:

$(-\infty, 2 - 2i) \cup (2 + 2i, \infty)$

Q: Why did we use the quadratic formula to find the roots of the quadratic equation $x^2 - 4x + 8 = 0$?

A: We used the quadratic formula to find the roots of the quadratic equation $x^2 - 4x + 8 = 0$ because the quadratic expression does not factor easily. The quadratic formula provides a general method for finding the roots of a quadratic equation.

Q: What is the significance of the roots $x = 2 \pm 2i$?

A: The roots $x = 2 \pm 2i$ are complex numbers, which means that the quadratic expression $x^2 - 4x + 8$ does not have any real roots. This is important because it affects the sign of the quadratic expression for different values of $x$.

Q: How did we understand the sign of the quadratic expression for different values of $x$?

A: We understood the sign of the quadratic expression for different values of $x$ by creating a sign chart or by graphing the quadratic function. This helped us visualize the intervals where the quadratic expression is positive, negative, or zero.

Q: What is the relationship between the sign chart and the graph of the quadratic function?

A: The sign chart and the graph of the quadratic function are related in that they both help us understand the sign of the quadratic expression for different values of $x$. The sign chart provides a more detailed and precise way of understanding the sign of the quadratic expression, while the graph of the quadratic function provides a visual representation of the sign of the quadratic expression.

Q: Why is it important to understand the sign of the quadratic expression for different values of $x$?

A: It is important to understand the sign of the quadratic expression for different values of $x$ because it helps us determine the solution to the inequality. By understanding the sign of the quadratic expression, we can identify the intervals where the quadratic expression is negative, which is the solution to the inequality.

Q: What is the final answer to the inequality $x^2 - 4x + 8 \leq 0$?

A: The final answer to the inequality $x^2 - 4x + 8 \leq 0$ is:

$(-\infty, 2 - 2i) \cup (2 + 2i, \infty)$

Conclusion

In this article, we provided a Q&A section to help clarify any doubts or questions that readers may have about solving the inequality $x^2 - 4x + 8 \leq 0$. We hope that this Q&A section has been helpful in understanding the solution to the inequality.

Final Answer

The final answer is: $\boxed{(-\infty, 2 - 2i) \cup (2 + 2i, \infty)}$