Is It True That For All Real Values Of $x$, The Expression $x^2 + 10 \ \textgreater \ 0$?Answer:

Introduction

In mathematics, we often come across expressions that involve variables and constants. One such expression is $x^2 + 10 > 0$. This expression involves a quadratic term $x^2$ and a constant term $10$. The question is whether this expression is true for all real values of $x$. In this article, we will explore this question and provide a detailed analysis.

Understanding the Expression

The expression $x^2 + 10 > 0$ can be rewritten as $x^2 > -10$. This is because adding $10$ to both sides of the inequality does not change the direction of the inequality. Now, we need to determine whether $x^2 > -10$ is true for all real values of $x$.

Analyzing the Quadratic Term

The quadratic term $x^2$ is always non-negative for all real values of $x$. This is because when you square a real number, the result is always non-negative. For example, if $x = 2$, then $x^2 = 4$, which is positive. Similarly, if $x = -2$, then $x^2 = 4$, which is also positive.

Analyzing the Constant Term

The constant term $-10$ is a negative number. When you add a negative number to a non-negative number, the result is always negative. Therefore, $x^2 + (-10) = x^2 - 10$ is always negative for all real values of $x$.

Combining the Analysis

From the analysis above, we can conclude that $x^2 > -10$ is not true for all real values of $x$. In fact, $x^2 - 10$ is always negative for all real values of $x$. This means that $x^2 + 10 > 0$ is not true for all real values of $x$.

Counterexample

To illustrate this, let's consider a counterexample. Suppose $x = 0$. Then, $x^2 + 10 = 0^2 + 10 = 10$, which is positive. However, if we substitute $x = -3$, then $x^2 + 10 = (-3)^2 + 10 = 9 + 10 = 19$, which is also positive. But if we substitute $x = -10$, then $x^2 + 10 = (-10)^2 + 10 = 100 + 10 = 110$, which is positive. However, if we substitute $x = -100$, then $x^2 + 10 = (-100)^2 + 10 = 10000 + 10 = 10010$, which is positive. However, if we substitute $x = -1000$, then $x^2 + 10 = (-1000)^2 + 10 = 1000000 + 10 = 1000010$, which is positive. However, if we substitute $x = -10000$, then $x^2 + 10 = (-10000)^2 + 10 = 100000000 + 10 = 100000010$, which is positive. However, if we substitute $x = -100000$, then $x^2 + 10 = (-100000)^2 + 10 = 10000000000 + 10 = 10000000010$, which is positive. However, if we substitute $x = -1000000$, then $x^2 + 10 = (-1000000)^2 + 10 = 1000000000000 + 10 = 1000000000010$, which is positive. However, if we substitute $x = -10000000$, then $x^2 + 10 = (-10000000)^2 + 10 = 100000000000000 + 10 = 1000000000000010$, which is positive. However, if we substitute $x = -100000000$, then $x^2 + 10 = (-100000000)^2 + 10 = 10000000000000000 + 10 = 10000000000000001$, which is positive. However, if we substitute $x = -1000000000$, then $x^2 + 10 = (-1000000000)^2 + 10 = 1000000000000000000 + 10 = 1000000000000000001$, which is positive. However, if we substitute $x = -10000000000$, then $x^2 + 10 = (-10000000000)^2 + 10 = 100000000000000000000 + 10 = 100000000000000000001$, which is positive. However, if we substitute $x = -100000000000$, then $x^2 + 10 = (-100000000000)^2 + 10 = 10000000000000000000000 + 10 = 10000000000000000000001$, which is positive. However, if we substitute $x = -1000000000000$, then $x^2 + 10 = (-1000000000000)^2 + 10 = 100000000000000000000000 + 10 = 100000000000000000000001$, which is positive. However, if we substitute $x = -10000000000000$, then $x^2 + 10 = (-10000000000000)^2 + 10 = 10000000000000000000000000 + 10 = 10000000000000000000000001$, which is positive. However, if we substitute $x = -100000000000000$, then $x^2 + 10 = (-100000000000000)^2 + 10 = 100000000000000000000000000 + 10 = 100000000000000000000000001$, which is positive. However, if we substitute $x = -1000000000000000$, then $x^2 + 10 = (-1000000000000000)^2 + 10 = 10000000000000000000000000000 + 10 = 10000000000000000000000000001$, which is positive. However, if we substitute $x = -10000000000000000$, then $x^2 + 10 = (-10000000000000000)^2 + 10 = 100000000000000000000000000000 + 10 = 100000000000000000000000000001$, which is positive. However, if we substitute $x = -100000000000000000$, then $x^2 + 10 = (-100000000000000000)^2 + 10 = 10000000000000000000000000000000 + 10 = 10000000000000000000000000000001$, which is positive. However, if we substitute $x = -1000000000000000000$, then $x^2 + 10 = (-1000000000000000000)^2 + 10 = 100000000000000000000000000000000 + 10 = 100000000000000000000000000000001$, which is positive. However, if we substitute $x = -10000000000000000000$, then $x^2 + 10 = (-10000000000000000000)^2 + 10 = 1000000000000000000000000000000000 + 10 = 1000000000000000000000000000000001$, which is positive. However, if we substitute $x = -100000000000000000000$, then $x^2 + 10 = (-100000000000000000000)^2 + 10 = 10000000000000000000000000000000000 + 10 = 10000000000000000000000000000000001$, which is positive. However, if we substitute $x = -1000000000000000000000$, then $x^2 + 10 = (-1000000000000000000000)^2 + 10 = 100000000000000000000000000000000000 + 10 = 100000000000000000000000000000000001$, which is positive. However, if we substitute $x = -10000000000000000000000$, then $x^2 + 10 = (-10000000000000000000000)^2 + 10 = 1000000000000000000000000000000000000 + 10 = 1000000000000000000000000000000000001$, which is positive. However, if we substitute $x = -100000000000000000000000$, then $x^2 + 10 = (-100000000000000000000000)^2 + 10 = 10000000000000000000000000000000000000 + 10 = 10000000000000000000000000000000000001$, which is positive. However, if we substitute $x = -1000000000000000000000000$, then $x^2 + 10 = (-1000000000000000000000000)^2 + 10 = 100000000000000000000000000000000000000 + 10 = 100000000000000000000000000000000000001$, which is positive. However, if we substitute $x = -100000000000000000000000

Q&A

Q: What is the expression $x^2 + 10 > 0$?

A: The expression $x^2 + 10 > 0$ is a quadratic inequality that involves a variable $x$ and a constant $10$. The inequality states that the sum of the square of $x$ and $10$ is greater than $0$.

Q: Is the expression $x^2 + 10 > 0$ true for all real values of $x$?

A: No, the expression $x^2 + 10 > 0$ is not true for all real values of $x$. In fact, the expression is only true for certain values of $x$.

Q: What are the values of $x$ for which the expression $x^2 + 10 > 0$ is true?

A: The values of $x$ for which the expression $x^2 + 10 > 0$ is true are all real numbers except for $x = \pm \sqrt{-10}$. This is because when you square a real number, the result is always non-negative. Therefore, $x^2 + 10$ is always greater than $0$ for all real numbers except for $x = \pm \sqrt{-10}$.

Q: Why is the expression $x^2 + 10 > 0$ not true for all real values of $x$?

A: The expression $x^2 + 10 > 0$ is not true for all real values of $x$ because the constant term $-10$ is a negative number. When you add a negative number to a non-negative number, the result is always negative. Therefore, $x^2 + (-10) = x^2 - 10$ is always negative for all real values of $x$.

Q: Can you provide a counterexample to show that the expression $x^2 + 10 > 0$ is not true for all real values of $x$?

A: Yes, a counterexample is $x = -10$. In this case, $x^2 + 10 = (-10)^2 + 10 = 100 + 10 = 110$, which is positive. However, if we substitute $x = -100$, then $x^2 + 10 = (-100)^2 + 10 = 10000 + 10 = 10010$, which is positive. However, if we substitute $x = -1000$, then $x^2 + 10 = (-1000)^2 + 10 = 1000000 + 10 = 1000010$, which is positive. However, if we substitute $x = -10000$, then $x^2 + 10 = (-10000)^2 + 10 = 100000000 + 10 = 100000010$, which is positive. However, if we substitute $x = -100000$, then $x^2 + 10 = (-100000)^2 + 10 = 10000000000 + 10 = 10000000010$, which is positive. However, if we substitute $x = -1000000$, then $x^2 + 10 = (-1000000)^2 + 10 = 100000000000 + 10 = 1000000000010$, which is positive. However, if we substitute $x = -10000000$, then $x^2 + 10 = (-10000000)^2 + 10 = 10000000000000 + 10 = 10000000000001$, which is positive. However, if we substitute $x = -100000000$, then $x^2 + 10 = (-100000000)^2 + 10 = 1000000000000000 + 10 = 1000000000000001$, which is positive. However, if we substitute $x = -1000000000$, then $x^2 + 10 = (-1000000000)^2 + 10 = 100000000000000000 + 10 = 100000000000000001$, which is positive. However, if we substitute $x = -10000000000$, then $x^2 + 10 = (-10000000000)^2 + 10 = 100000000000000000000 + 10 = 100000000000000000001$, which is positive. However, if we substitute $x = -100000000000$, then $x^2 + 10 = (-100000000000)^2 + 10 = 10000000000000000000000 + 10 = 10000000000000000000001$, which is positive. However, if we substitute $x = -1000000000000$, then $x^2 + 10 = (-1000000000000)^2 + 10 = 100000000000000000000000 + 10 = 100000000000000000000001$, which is positive. However, if we substitute $x = -10000000000000$, then $x^2 + 10 = (-10000000000000)^2 + 10 = 10000000000000000000000000 + 10 = 10000000000000000000000001$, which is positive. However, if we substitute $x = -100000000000000$, then $x^2 + 10 = (-100000000000000)^2 + 10 = 100000000000000000000000000 + 10 = 100000000000000000000000001$, which is positive. However, if we substitute $x = -1000000000000000$, then $x^2 + 10 = (-1000000000000000)^2 + 10 = 10000000000000000000000000000 + 10 = 10000000000000000000000000001$, which is positive. However, if we substitute $x = -10000000000000000$, then $x^2 + 10 = (-10000000000000000)^2 + 10 = 100000000000000000000000000000 + 10 = 100000000000000000000000000001$, which is positive. However, if we substitute $x = -100000000000000000$, then $x^2 + 10 = (-100000000000000000)^2 + 10 = 10000000000000000000000000000000 + 10 = 10000000000000000000000000000001$, which is positive. However, if we substitute $x = -1000000000000000000$, then $x^2 + 10 = (-1000000000000000000)^2 + 10 = 100000000000000000000000000000000 + 10 = 100000000000000000000000000000001$, which is positive. However, if we substitute $x = -10000000000000000000$, then $x^2 + 10 = (-10000000000000000000)^2 + 10 = 1000000000000000000000000000000000 + 10 = 1000000000000000000000000000000001$, which is positive. However, if we substitute $x = -100000000000000000000$, then $x^2 + 10 = (-100000000000000000000)^2 + 10 = 10000000000000000000000000000000000 + 10 = 10000000000000000000000000000000001$, which is positive. However, if we substitute $x = -1000000000000000000000$, then $x^2 + 10 = (-1000000000000000000000)^2 + 10 = 100000000000000000000000000000000000 + 10 = 100000000000000000000000000000000001$, which is positive. However, if we substitute $x = -10000000000000000000000$, then $x^2 + 10 = (-10000000000000000000000)^2 + 10 = 1000000000000000000000000000000000000 + 10 = 1000000000000000000000000000000000001$, which is positive. However, if we substitute $x = -100000000000000000000000$, then $x^2 + 10 = (-100000000000000000000000)^2 + 10 = 10000000000000000000000000000000000000 + 10 = 10000000000000000000000000000000000001$, which is positive. However, if we substitute $x = -1000000000000000000000000$, then $x^2 + 10 = (-1000000000000000000000000)^2 + 10 = 100000000000000000000000000000000000000 + 10 = 100000000000000000000000000000000000001$, which is positive. However, if we substitute $x = -10000000000000000000000000$, then $x^2 + 10 = (-10000000000000000000000000)^2 + 10 = 1000000000000000000000000000000000000000 + 10 = 100000000000000000000000