Solve y=-3x²-9: Finding Values Where Function is Positive

Q: Why can't x² ever be negative?

Because when you square any real number, positive or negative, the result is always positive or zero. For example: $ 5^2 = 25 $ and $ (-5)^2 = 25 $.

Q: What does it mean when there are no solutions?

It means the function $ y = -3x^2 - 9 $ is never positive for any real value of x. The parabola opens downward and its highest point is at $ y = -9 $.

Q: How do I know when to flip the inequality sign?

Always flip when multiplying or dividing both sides by a negative number. In this problem, dividing by -3 changes $ -3x^2 > 9 $ to \( x^2 .

Q: Could the answer be 'all values of x' instead?

No! The function $ y = -3x^2 - 9 $ has a maximum value of -9 (when x = 0). Since -9 never positive.

Q: What would the graph of this function look like?

It's a downward-opening parabola with vertex at (0, -9). The entire graph lies below the x-axis, which confirms that $ f(x) > 0 $ has no solutions.

Quadratic Inequalities with No Solutions

Look at the function below:

$y=-3x^2-9$

Determine for which values of $x$ the following is true:

$f(x) > 0$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the function below:

$y=-3x^2-9$

Determine for which values of $x$ the following is true:

$f(x) > 0$

Step-by-step solution

To solve this problem, let's analyze the function $f(x) = -3x^2 - 9$ and determine when it is greater than zero.

The function is a quadratic equation of the form $ax^2 + bx + c$ where $a = -3$ , $b = 0$ , and $c = -9$ .

Our task is to find when $-3x^2 - 9 > 0$ .

Step 1: Rewrite the inequality:
$-3x^2 - 9 > 0$

Step 2: Add 9 to both sides to isolate the quadratic term:
$-3x^2 > 9$

Step 3: Divide through by -3 (note that dividing by a negative flips the inequality sign):
$x^2 < -3$

Step 4: Analyze $x^2 < -3$

A square of a real number $x^2$ is always non-negative, meaning $x^2 \geq 0$ . Therefore, $x^2 < -3$ is impossible since there are no real values of $x$ such that the square of $x$ is a negative number.

Conclusion: The inequality $f(x) > 0$ has no real solutions. Therefore, no values of $x$ satisfy the inequality.

The correct answer is that no values of $x$ will make $f(x) > 0$ .

Final Answer

No values of $x$

Key Points to Remember

Essential concepts to master this topic

Rule: Square of any real number is always non-negative
Technique: Isolate x² term: -3x² > 9 becomes x² < -3
Check: Since x² ≥ 0 always, x² < -3 is impossible ✓

Common Mistakes

Avoid these frequent errors

Forgetting to flip inequality sign when dividing by negative
Don't divide -3x² > 9 by -3 and keep the > sign = x² > -3! This gives the wrong inequality direction and leads to incorrect analysis. Always flip the inequality sign when multiplying or dividing by negative numbers.

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 can't x² ever be negative?

Because when you square any real number, positive or negative, the result is always positive or zero. For example: $5^2 = 25$ and $(-5)^2 = 25$ .

What does it mean when there are no solutions?

It means the function $y = -3x^2 - 9$ is never positive for any real value of x. The parabola opens downward and its highest point is at $y = -9$ .

How do I know when to flip the inequality sign?

Always flip when multiplying or dividing both sides by a negative number. In this problem, dividing by -3 changes $-3x^2 > 9$ to $x^2 < -3$ .

Could the answer be 'all values of x' instead?

No! The function $y = -3x^2 - 9$ has a maximum value of -9 (when x = 0). Since -9 < 0, the function is never positive.

What would the graph of this function look like?

It's a downward-opening parabola with vertex at (0, -9). The entire graph lies below the x-axis, which confirms that $f(x) > 0$ has no solutions.

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