Solve the Quadratic Inequality: -x² + 4x > 0

Q: Why do I need to find where the expression equals zero first?

The zeros (roots) are critical points that divide the number line into intervals. The expression can only change sign at these points, so they help us determine where the inequality is satisfied.

Q: How do I know which intervals to test?

After finding zeros at x = 0 and x = 4, test any point from each interval: $(-∞, 0)$, $(0, 4)$, and $(4, ∞)$. Pick easy numbers like x = -1, x = 2, and x = 5.

Q: What's the difference between > and ≥ in the final answer?

Strict inequality (>) means boundary points are excluded - use open intervals like (0, 4). Non-strict inequality (≥) includes boundary points - use closed intervals like [0, 4].

Q: Can I solve this by graphing instead?

Yes! Graph $y = -x^2 + 4x$ and find where the parabola is above the x-axis (y > 0). This visually shows the solution interval (0, 4).

Quadratic Inequalities with Factoring Methods

Solve the following equation:

$-x^2+4x>0$

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Understand the problem

Solve the following equation:

$-x^2+4x>0$

Step-by-step solution

To understand how to solve the inequality $-x^2 + 4x > 0$ , let's break it down step-by-step:

Step 1: Solve the related quadratic equation. Begin with $-x^2 + 4x = 0$ . We can factor this as $x(-x + 4) = 0$ , giving roots $x = 0$ and $x = 4$ .
Step 2: Use these roots to divide the real number line into intervals: $(-∞, 0)$ , $(0, 4)$ , and $(4, ∞)$ .
Step 3: Test a point from each interval in the inequality:

For $(-∞, 0)$ , choose $x = -1$ : $-(-1)^2 + 4(-1) = -1 - 4 = -5$ (negative)
For $(0, 4)$ , choose $x = 2$ : $-(2)^2 + 4(2) = -4 + 8 = 4$ (positive)
For $(4, ∞)$ , choose $x = 5$ : $-(5)^2 + 4(5) = -25 + 20 = -5$ (negative)

Step 4: Identify where $-x^2 + 4x > 0$ : Only in the interval $(0, 4)$ .

Therefore, the solution to the inequality $-x^2 + 4x > 0$ is the interval $(0, 4)$ . This corresponds to the choice: $0 < x < 4$ .

Final Answer

$0 < x < 4$

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Factor completely before finding critical points from equality
Sign Testing: Test points from each interval: x = 2 gives 4 > 0
Verification: Boundary points x = 0 and x = 4 make expression equal zero ✓

Common Mistakes

Avoid these frequent errors

Including boundary points in strict inequality solutions
Don't write 0 ≤ x ≤ 4 for -x² + 4x > 0 = wrong solution set! The inequality is strict (>) not (≥), so boundary points where the expression equals zero must be excluded. Always use open intervals (0, 4) for strict inequalities.

Practice Quiz

Test your knowledge with interactive questions

Solve the following equation:

$$ x^2+4>0 $$

FAQ

Everything you need to know about this question

Why do I need to find where the expression equals zero first?

The zeros (roots) are critical points that divide the number line into intervals. The expression can only change sign at these points, so they help us determine where the inequality is satisfied.

How do I know which intervals to test?

After finding zeros at x = 0 and x = 4, test any point from each interval: $(-∞, 0)$ , $(0, 4)$ , and $(4, ∞)$ . Pick easy numbers like x = -1, x = 2, and x = 5.

What's the difference between > and ≥ in the final answer?

Strict inequality (>) means boundary points are excluded - use open intervals like (0, 4). Non-strict inequality (≥) includes boundary points - use closed intervals like [0, 4].

Can I solve this by graphing instead?

Yes! Graph $y = -x^2 + 4x$ and find where the parabola is above the x-axis (y > 0). This visually shows the solution interval (0, 4).

Why is my sign analysis showing different results?

Double-check your arithmetic when substituting test points. For example, at x = 2: $-(2)^2 + 4(2) = -4 + 8 = 4$ , which is positive. Always verify your calculations!

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