Why is the answer 'x 6' and not 'and'?

We want all x-values where f(x) 'or' to include both regions.

Solve the Quadratic Inequality y = -4x² + 24x: Which x-values Apply?

Q: Why do we set the function equal to zero first?

Setting $ f(x) = 0 $ finds the roots where the parabola crosses the x-axis. These roots divide the number line into intervals where the function stays either positive or negative.

Q: How do I know which intervals to test?

The roots create boundary points that split the x-axis. For roots at x = 0 and x = 6, test intervals: (-∞, 0), (0, 6), and (6, ∞).

Q: What if I get the wrong sign when testing?

Double-check your arithmetic! For x = -1: $ -4(-1)^2 + 24(-1) = -4(1) - 24 = -28 $. Remember that (-1)² = 1, not -1.

Q: Can I solve this without graphing?

Yes! The sign analysis method (finding roots, creating intervals, testing points) works perfectly without graphing. It's actually more reliable than sketching!

Quadratic Inequalities with Sign Analysis

Look at the following function:

$y=-4x^2+24x$

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 following function:

$y=-4x^2+24x$

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

$f(x) < 0$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Find the roots by setting the function equal to zero: $-4x^2 + 24x = 0$ .
Step 2: Factor the equation. We get $-4x(x - 6) = 0$ .
Step 3: Solve for $x$ to find the roots: $x = 0$ and $x = 6$ .
Step 4: Analyze intervals defined by these roots: these intervals are $(-\infty, 0)$ , $(0, 6)$ , and $(6, \infty)$ .
Step 5: Test the sign of $y = -4x^2 + 24x$ in each interval:

In $(-\infty, 0)$ , choose $x = -1$ : $-4(-1)^2 + 24(-1) = -4 - 24 = -28$ , so $y < 0$ .
In $(0, 6)$ , choose $x = 3$ : $-4(3)^2 + 24(3) = -36 + 72 = 36$ , so $y > 0$ .
In $(6, \infty)$ , choose $x = 7$ : $-4(7)^2 + 24(7) = -196 + 168 = -28$ , so $y < 0$ .

Step 6: Compile the solution set based on where $y < 0$ : the intervals are $(-\infty, 0)$ and $(6, \infty)$ .

Therefore, the solution is that $f(x) < 0$ for $x > 6$ or $x < 0$ .

Final Answer

$x > 6$ or $x < 0$

Key Points to Remember

Essential concepts to master this topic

Factoring: Factor out common terms to find roots easily
Test Points: Choose x = -1, 3, 7 in intervals (-∞,0), (0,6), (6,∞)
Verification: Test x = -1: -4(-1)² + 24(-1) = -28 < 0 ✓

Common Mistakes

Avoid these frequent errors

Testing only one interval instead of all three
Don't just test one interval and assume the pattern = wrong solution! The parabola opens downward, so signs change across roots. Always test a point in each interval created by the roots.

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below intersects the X-axis at points A and B.

The vertex of the parabola is marked at point C.

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

FAQ

Everything you need to know about this question

Why do we set the function equal to zero first?

Setting $f(x) = 0$ finds the roots where the parabola crosses the x-axis. These roots divide the number line into intervals where the function stays either positive or negative.

How do I know which intervals to test?

The roots create boundary points that split the x-axis. For roots at x = 0 and x = 6, test intervals: (-∞, 0), (0, 6), and (6, ∞).

What if I get the wrong sign when testing?

Double-check your arithmetic! For x = -1: $-4(-1)^2 + 24(-1) = -4(1) - 24 = -28$ . Remember that (-1)² = 1, not -1.

Why is the answer 'x < 0 or x > 6' and not 'and'?

We want all x-values where f(x) < 0. Since the function is negative in two separate intervals, we use 'or' to include both regions.

Can I solve this without graphing?

Yes! The sign analysis method (finding roots, creating intervals, testing points) works perfectly without graphing. It's actually more reliable than sketching!

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