Solve the Quadratic Inequality: x²-6x+9 < 0

Q: Why doesn't this inequality have any solutions?

Because $ (x-3)^2 $ is a perfect square, and squares of real numbers are always zero or positive. Since we need it to be less than zero, there's no real number that works!

Q: What does it mean when a quadratic has no solution?

It means the parabola never goes below the x-axis. In this case, $ y = (x-3)^2 $ just touches the x-axis at x = 3 but stays above or on it everywhere else.

Q: How can I recognize when an inequality has no solution?

Look for these clues: perfect square trinomials like $ (x-3)^2 $ are never negative, and upward-opening parabolas that touch but don't cross the x-axis.

Q: Is there ever a case where the inequality would have solutions?

Yes! If the inequality were $ (x-3)^2 > 0 $, then all real numbers except x = 3 would be solutions. The direction of the inequality matters!

Q: What's the difference between < 0 and ≤ 0 for this problem?

No solutions (squares can't be negative)≤ 0: Only x = 3 works (when the square equals zero)

Perfect Square Trinomials with No Solution

Solve the following equation:

$x^2-6x+9<0$

❤️ 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:

$x^2-6x+9<0$

Step-by-step solution

To solve the inequality $x^2 - 6x + 9 < 0$ , we first factor the quadratic expression as $(x-3)^2$ .

Note that $(x-3)^2 \geq 0$ for all real numbers $x$ because it is a square. Furthermore, $(x-3)^2$ equals zero only when $x = 3$ .

This means that the expression $(x-3)^2$ never actually becomes negative for any real $x$ . The vertex of the parabola, a perfect square, simply touches the x-axis at $x = 3$ but does not dip below.

Therefore, there is no solution to the inequality $x^2 - 6x + 9 < 0$ over the real numbers.

The conclusion is that there is no negative domain.

Final Answer

There is no negative domain.

Key Points to Remember

Essential concepts to master this topic

Rule: Factor quadratic expressions first to reveal their structure
Technique: Recognize $x^2 - 6x + 9 = (x-3)^2$ as perfect square
Check: Squares are always ≥ 0, so (x-3)² < 0 is impossible ✓

Common Mistakes

Avoid these frequent errors

Assuming all quadratic inequalities have solutions
Don't try to solve (x-3)² < 0 by finding x-values = wrong approach! Perfect squares can never be negative. Always check if the factored form makes the inequality mathematically possible.

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 doesn't this inequality have any solutions?

Because $(x-3)^2$ is a perfect square, and squares of real numbers are always zero or positive. Since we need it to be less than zero, there's no real number that works!

What does it mean when a quadratic has no solution?

It means the parabola never goes below the x-axis. In this case, $y = (x-3)^2$ just touches the x-axis at x = 3 but stays above or on it everywhere else.

How can I recognize when an inequality has no solution?

Look for these clues: perfect square trinomials like $(x-3)^2$ are never negative, and upward-opening parabolas that touch but don't cross the x-axis.

Is there ever a case where the inequality would have solutions?

Yes! If the inequality were $(x-3)^2 > 0$ , then all real numbers except x = 3 would be solutions. The direction of the inequality matters!

What's the difference between < 0 and ≤ 0 for this problem?

< 0: No solutions (squares can't be negative)
≤ 0: Only x = 3 works (when the square equals zero)

🌟 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