Finding X in the Quadratic Inequality: Where y = -x² < 0

Q: Why is the answer x ≠ 0 instead of all x values?

Great question! At $ x = 0 $, we get $ y = -(0)^2 = 0 $. Since we want f(x) (strictly less than zero), we must exclude x = 0 where the function equals zero.

Q: How do I know the parabola opens downward?

Look at the coefficient of $ x^2 $! Since we have $ -x^2 $, the coefficient is -1 (negative). Negative coefficients make parabolas open downward, while positive coefficients make them open upward.

Q: What if the inequality was f(x) ≤ 0 instead?

Then the answer would be all real numbers! The ≤ symbol means "less than or equal to," so we'd include x = 0 where the function equals zero.

Q: Can I solve this by factoring like other quadratic inequalities?

You could write \( -x^2 as \( -x \cdot x , but it's easier to recognize that any negative number squared gives a negative result when multiplied by -1.

Q: How do I check my answer is correct?

Test a few values! Try x = 2: $ -(2)^2 = -4 . Try x = -3: \( -(-3)^2 = -9 . Try x = 0: \( -(0)^2 = 0 $ (not

Quadratic Inequalities with Downward-Opening Parabolas

Given the function:

$y=-x^2$

Determine for which values of x $f(x) < 0$ holds

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the function:

$y=-x^2$

Determine for which values of x $f(x) < 0$ holds

Step-by-step solution

To solve this problem, we'll analyze the quadratic function $y = -x^2$ .

Step 1: Determine when $y = -x^2$ is less than zero.
The function value is zero at $x = 0$ and less than zero for any other $x$ . Since $-x^2$ will be negative for all non-zero $x$ , this means $f(x) < 0$ for all $x \ne 0$ .
Step 2: Interpretation of results.
The graph of $y = -x^2$ opens downward and only touches the x-axis at $x = 0$ . This means, except at the point where $x = 0$ , the function outputs negative values.
Step 3: Comparison with given choices.
We want $f(x) < 0$ . Since this holds for all $x \ne 0$ , we will select the answer $x \ne 0$ .

Therefore, the solution to the problem is $x \ne 0$ .

Final Answer

$x\ne0$

Key Points to Remember

Essential concepts to master this topic

Rule: For $y = -x^2$ , the parabola opens downward and equals zero at x = 0
Technique: Test values: at x = 1, $y = -(1)^2 = -1 < 0$
Check: Graph shows function is negative everywhere except x = 0 where y = 0 ✓

Common Mistakes

Avoid these frequent errors

Thinking negative coefficients make all outputs positive
Don't assume $-x^2$ gives positive values = wrong inequality direction! The negative sign applies to the squared term, making outputs negative or zero. Always substitute test values to verify the sign of function outputs.

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 is the answer x ≠ 0 instead of all x values?

Great question! At $x = 0$ , we get $y = -(0)^2 = 0$ . Since we want f(x) < 0 (strictly less than zero), we must exclude x = 0 where the function equals zero.

How do I know the parabola opens downward?

Look at the coefficient of $x^2$ ! Since we have $-x^2$ , the coefficient is -1 (negative). Negative coefficients make parabolas open downward, while positive coefficients make them open upward.

What if the inequality was f(x) ≤ 0 instead?

Then the answer would be all real numbers! The ≤ symbol means "less than or equal to," so we'd include x = 0 where the function equals zero.

Can I solve this by factoring like other quadratic inequalities?

You could write $-x^2 < 0$ as $-x \cdot x < 0$ , but it's easier to recognize that any negative number squared gives a negative result when multiplied by -1.

How do I check my answer is correct?

Test a few values! Try x = 2: $-(2)^2 = -4 < 0 ✓$ . Try x = -3: $-(-3)^2 = -9 < 0 ✓$ . Try x = 0: $-(0)^2 = 0$ (not < 0).

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