Solve x² + 4 > 0: Understanding Quadratic Inequalities

Q: Why can't I solve x² + 4 = 0 to find boundary points?

Because $ x^2 + 4 = 0 $ means $ x^2 = -4 $, but squares of real numbers cannot be negative! This equation has no real solutions, so there are no boundary points to consider.

Q: How do I know when a quadratic expression is always positive?

Look for expressions like $ x^2 + \text{positive number} $. Since $ x^2 \geq 0 $ always, adding a positive constant makes the entire expression always positive.

Q: What if the problem was x² - 4 > 0 instead?

Then you would solve $ x^2 - 4 = 0 $ to find boundary points x = ±2, then test intervals. The key difference is subtracting vs. adding the constant!

Q: Do I need to graph this inequality?

Not necessary here! Since $ x^2 + 4 $ is always positive, the graph of y = x² + 4 is always above the x-axis. The solution is simply all real numbers.

Quadratic Inequalities with Always-Positive Expressions

Solve the following equation:

$x^2+4>0$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

$x^2+4>0$

Step-by-step solution

To solve this problem, let's examine the inequality $x^2 + 4 > 0$ .

The expression $x^2 + 4$ consists of two terms: $x^2$ and $4$ . Notice that:

The term $x^2$ is always non-negative, which means $x^2 \geq 0$ for any real number $x$ .
The constant term $4$ is positive.

Combining these observations, we see that:

Since $x^2$ is non-negative, $x^2 + 4 \geq 4$ .
Therefore, $x^2 + 4$ is always greater than zero, as adding 4 to a non-negative number will always yield a positive result.

Thus, there are no values of $x$ for which the expression $x^2 + 4$ is zero or negative. Instead, the expression is always positive for all real numbers $x$ .

Therefore, the solution to the inequality $x^2 + 4 > 0$ is all values of $x$ .

Final Answer

All values of $x$

Key Points to Remember

Essential concepts to master this topic

Property: x² is always non-negative for any real number x
Analysis: Since x² ≥ 0, then x² + 4 ≥ 4 > 0
Verification: Test any value like x = 0: 0² + 4 = 4 > 0 ✓

Common Mistakes

Avoid these frequent errors

Trying to solve x² + 4 = 0 first
Don't attempt to find roots by setting x² + 4 = 0 since this has no real solutions! This wastes time and confuses the analysis. Always recognize that x² + positive constant is always positive for all real numbers.

Practice Quiz

Test your knowledge with interactive questions

Solve the following equation:

$$ x^2+4>0 $$

FAQ

Everything you need to know about this question

Why can't I solve x² + 4 = 0 to find boundary points?

Because $x^2 + 4 = 0$ means $x^2 = -4$ , but squares of real numbers cannot be negative! This equation has no real solutions, so there are no boundary points to consider.

How do I know when a quadratic expression is always positive?

Look for expressions like $x^2 + \text{positive number}$ . Since $x^2 \geq 0$ always, adding a positive constant makes the entire expression always positive.

What if the problem was x² - 4 > 0 instead?

Then you would solve $x^2 - 4 = 0$ to find boundary points x = ±2, then test intervals. The key difference is subtracting vs. adding the constant!

Do I need to graph this inequality?

Not necessary here! Since $x^2 + 4$ is always positive, the graph of y = x² + 4 is always above the x-axis. The solution is simply all real numbers.

How can I be sure my answer covers all real numbers?

Test several values: when x = 0, we get 4 > 0 ✓. When x = -3, we get 9 + 4 = 13 > 0 ✓. When x = 100, we get 10,004 > 0 ✓. Every test confirms the pattern!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Equations and Systems of Quadratic Equations questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime