Finding Solutions When x² + 4x is Positive

Q: Why do I need to test points in each interval?

Testing points tells you the sign of the expression in each region! Since quadratics are continuous, if one point in an interval is positive, the whole interval will be positive.

Q: What happens at the boundary points x = -4 and x = 0?

At these points, the expression $ x^2 + 4x = 0 $, which is not positive. Since we need the expression to be greater than zero, we exclude these boundary points.

Q: How do I remember which intervals to include?

Make a sign chart! Draw a number line, mark your roots, then test one point in each interval. Include intervals where your test gives a positive result.

Q: Can I solve this by graphing instead?

Absolutely! Graph $ y = x^2 + 4x $ and find where the parabola is above the x-axis. This happens when $ x or \( x > 0 $.

Q: What if I get confused about the direction of the inequality?

Remember: we want $ x^2 + 4x > 0 $, meaning the expression is positive. Choose intervals where your test points give positive results, not negative ones!

Quadratic Inequalities with Sign Analysis

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 solve the quadratic inequality $x^2 + 4x > 0$ , we follow these steps:

Step 1: Start by solving the quadratic equation $x^2 + 4x = 0$ to find its roots. Factoring, we get:

$x(x + 4) = 0$

This gives roots $x = 0$ and $x = -4$ .

Step 2: Use these roots to split the number line into intervals to test the sign of $x(x + 4)$ within each interval. The intervals are:
Interval 1: $x < -4$
Interval 2: $-4 < x < 0$
Interval 3: $x > 0$

Step 3: Choose a test point from each interval to determine the sign of $x(x + 4)$ . For instance:
Test point for Interval 1: $x = -5$ . We have $(-5)((-5) + 4) = 5$ , which is positive.
Test point for Interval 2: $x = -2$ . We have $(-2)((-2) + 4) = -4$ , which is negative.
Test point for Interval 3: $x = 1$ . We have $1(1 + 4) = 5$ , which is positive.

Step 4: Compile the results. The inequality $x(x + 4) > 0$ holds for:
$x < -4$ (Interval 1)
$x > 0$ (Interval 3)

Thus the solution to the inequality $x^2 + 4x > 0$ is:

$x < -4$ or $x > 0$

Final Answer

$x < -4,0 < x$

Key Points to Remember

Essential concepts to master this topic

Rule: Factor the quadratic and find roots first
Technique: Test signs in intervals: x = -5 gives (-5)(-1) = 5 > 0
Check: Verify boundary points aren't included: x = 0 and x = -4 make expression equal zero, not positive ✓

Common Mistakes

Avoid these frequent errors

Including the boundary points in the solution
Don't write x ≤ -4 or x ≥ 0 for a strict inequality = wrong solution set! The expression equals zero at x = -4 and x = 0, but we need it to be positive (greater than zero). Always use strict inequalities < and > when the original inequality is strict.

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 test points in each interval?

Testing points tells you the sign of the expression in each region! Since quadratics are continuous, if one point in an interval is positive, the whole interval will be positive.

What happens at the boundary points x = -4 and x = 0?

At these points, the expression $x^2 + 4x = 0$ , which is not positive. Since we need the expression to be greater than zero, we exclude these boundary points.

How do I remember which intervals to include?

Make a sign chart! Draw a number line, mark your roots, then test one point in each interval. Include intervals where your test gives a positive result.

Can I solve this by graphing instead?

Absolutely! Graph $y = x^2 + 4x$ and find where the parabola is above the x-axis. This happens when $x < -4$ or $x > 0$ .

What if I get confused about the direction of the inequality?

Remember: we want $x^2 + 4x > 0$ , meaning the expression is positive. Choose intervals where your test points give positive results, not negative ones!

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