Solve Quadratic Inequality: When is -3x² - 9 Less Than Zero

Q: Why doesn't this parabola cross the x-axis?

The equation $ -3x^2 - 9 = 0 $ gives us $ x^2 = -3 $. Since no real number squared equals a negative value, there are no real solutions and no x-intercepts!

Q: How do I know if the parabola is above or below the x-axis?

Test any point! Try $ x = 0 $: $ f(0) = -3(0)^2 - 9 = -9 $. Since this is negative and the parabola doesn't cross the x-axis, it's entirely below the x-axis.

Q: What if the coefficient of x² was positive instead?

If we had $ f(x) = 3x^2 + 9 $, the parabola would open upward with no real roots, meaning it's entirely above the x-axis. Then \( f(x) would have no solutions!

Q: Can I factor this function?

Yes! Factor out -3: $ f(x) = -3(x^2 + 3) $. Since $ x^2 + 3 $ is always positive (minimum value is 3), and we multiply by -3, the result is always negative.

Q: What does 'All x' mean as an answer?

'All x' means every real number satisfies the inequality. You can write this as $ x \in \mathbb{R} $ or \( -\infty in interval notation.

Quadratic Inequalities with No Real Roots

Given the function:

$y=-3x^2-9$

Determine for which values of x the following holds:

$f\left(x\right) < 0$

❤️ Continue Your Math Journey!

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

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the function:

$y=-3x^2-9$

Determine for which values of x the following holds:

$f\left(x\right) < 0$

Step-by-step solution

Given the quadratic function $f(x) = -3x^2 - 9$ , we want to determine when $f(x) < 0$ .

First, observe that the function is a downward-opening parabola because the coefficient of $x^2$ is negative ( $-3$ ). This means the parabola opens downwards.

To find when the parabola is below the x-axis ( $f(x) < 0$ ), we should first check whether there are any real roots, since this implies crossing the x-axis.

The function $f(x) = -3x^2 - 9$ is reformulated as:

$0 = -3x^2 - 9$ .

To find the roots, rearrange and solve:

$-3x^2 = 9$ or $x^2 = -3$ .

Since $x^2 = -3$ yields no real solutions (as no real number squared equals a negative), there are no x-intercepts.

This indicates the parabola does not cross the x-axis and is entirely below it (since it opens downward and has no real roots).

Thus, the function $f(x) < 0$ for all real values of $x$ .

Therefore, the condition $f(x) < 0$ is satisfied for all x, meaning the correct choice is:

All x

Final Answer

All x

Key Points to Remember

Essential concepts to master this topic

Discriminant: When b² - 4ac < 0, parabola has no x-intercepts
Analysis: Factor out -3 to get -3(x² + 3) < 0
Verification: Check any x-value: f(0) = -9 < 0 ✓

Common Mistakes

Avoid these frequent errors

Setting function equal to zero to solve inequality
Don't solve -3x² - 9 = 0 thinking it gives inequality solutions = wrong approach! This finds x-intercepts, not where function is negative. Always analyze the parabola's position relative to x-axis using discriminant and coefficient signs.

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below does not intersect the $$ x $$ -axis.

The parabola's vertex is marked A.

Find all values of $$ x $$ where
$f\left(x\right) > 0$ .

FAQ

Everything you need to know about this question

Why doesn't this parabola cross the x-axis?

The equation $-3x^2 - 9 = 0$ gives us $x^2 = -3$ . Since no real number squared equals a negative value, there are no real solutions and no x-intercepts!

How do I know if the parabola is above or below the x-axis?

Test any point! Try $x = 0$ : $f(0) = -3(0)^2 - 9 = -9$ . Since this is negative and the parabola doesn't cross the x-axis, it's entirely below the x-axis.

What if the coefficient of x² was positive instead?

If we had $f(x) = 3x^2 + 9$ , the parabola would open upward with no real roots, meaning it's entirely above the x-axis. Then $f(x) < 0$ would have no solutions!

Can I factor this function?

Yes! Factor out -3: $f(x) = -3(x^2 + 3)$ . Since $x^2 + 3$ is always positive (minimum value is 3), and we multiply by -3, the result is always negative.

What does 'All x' mean as an answer?

'All x' means every real number satisfies the inequality. You can write this as $x \in \mathbb{R}$ or $-\infty < x < \infty$ in interval notation.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 The Quadratic Function 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