Solve the Parabola Inequality: y = -x²- 5x When f(x) < 0

Q: Why do I need to find the roots first?

The roots are where the parabola crosses the x-axis, changing from positive to negative (or vice versa). These are the boundary points that divide the number line into intervals with consistent signs.

Q: How do I know which intervals to test?

The roots $ x = -5 $ and $ x = 0 $ create three intervals: $ x , \( -5 , and \( x > 0 $. Test one point from each interval.

Q: What if I get confused about the parabola direction?

Since the coefficient of $ x^2 $ is negative (-1), this parabola opens downward. But don't rely on this - always test points to be sure!

Q: Can I use a number line to visualize this?

Yes! Draw a number line, mark the roots at -5 and 0, then test points in each region. Mark + or - above each interval based on your test results.

Q: Why isn't the answer just between the roots?

That's a common mistake! Since this parabola opens downward, it's positive between the roots and negative outside them. Always verify with actual calculations.

Q: Do I include the roots x = -5 and x = 0 in my answer?

No! The question asks for $ f(x) (strictly less than), so the roots where \( f(x) = 0 $ are not included.

Quadratic Inequalities with Sign Analysis

Look at the following function:

$y=-x^2-5x$

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

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

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the following function:

$y=-x^2-5x$

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

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

Step-by-step solution

To determine where the function $y = -x^2 - 5x$ is negative, follow these steps:

Step 1: Find the roots of the equation $-x^2 - 5x = 0$ .
Step 2: Factor the equation: $-x(x + 5) = 0$ .
Step 3: Solve for $x$ :
- $x = 0$
- $x = -5$
Step 4: Analyze the sign of the quadratic function in the intervals determined by the roots:
- Interval 1: $x < -5$
- Interval 2: $-5 < x < 0$
- Interval 3: $x > 0$

Consider an example point in each interval to determine if $y$ is negative:

For $x < -5$ , choose $x = -6$ :
$y = -(-6)^2 - 5(-6) = -36 + 30 = -6$ , which is negative.
For $-5 < x < 0$ , choose $x = -1$ :
$y = -(-1)^2 - 5(-1) = -1 + 5 = 4$ , which is positive.
For $x > 0$ , choose $x = 1$ :
$y = -(1)^2 - 5(1) = -1 - 5 = -6$ , which is negative.

Therefore, the intervals where $f(x) < 0$ are $x < -5$ and $x > 0$ .

The correct answer is $x > 0$ or $x < -5$ .

Final Answer

$x > 0$ or $x < -5$

Key Points to Remember

Essential concepts to master this topic

Root Finding: Factor the quadratic to find where it equals zero
Sign Testing: Choose test points like x = -6, -1, 1 in each interval
Verification: Check that f(-6) = -6 and f(1) = -6 are both negative ✓

Common Mistakes

Avoid these frequent errors

Not testing all three intervals between the roots
Don't assume the parabola is negative everywhere just because it opens downward = missing positive regions! The sign changes at each root. Always test a point in every interval: x < -5, -5 < x < 0, and x > 0.

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 do I need to find the roots first?

The roots are where the parabola crosses the x-axis, changing from positive to negative (or vice versa). These are the boundary points that divide the number line into intervals with consistent signs.

How do I know which intervals to test?

The roots $x = -5$ and $x = 0$ create three intervals: $x < -5$ , $-5 < x < 0$ , and $x > 0$ . Test one point from each interval.

What if I get confused about the parabola direction?

Since the coefficient of $x^2$ is negative (-1), this parabola opens downward. But don't rely on this - always test points to be sure!

Can I use a number line to visualize this?

Yes! Draw a number line, mark the roots at -5 and 0, then test points in each region. Mark + or - above each interval based on your test results.

Why isn't the answer just between the roots?

That's a common mistake! Since this parabola opens downward, it's positive between the roots and negative outside them. Always verify with actual calculations.

Do I include the roots x = -5 and x = 0 in my answer?

No! The question asks for $f(x) < 0$ (strictly less than), so the roots where $f(x) = 0$ are not included.

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