Solve the Quadratic Inequality: 3x² + 6x > 0

Q: Why do I need to factor the expression first?

Factoring $ 3x^2 + 6x = 3x(x + 2) $ reveals the critical points where the function equals zero. These points divide the number line into intervals where the sign stays constant!

Q: How do I know which intervals to test?

The roots $ x = -2 $ and $ x = 0 $ create three intervals: $ (-\infty, -2) $, $ (-2, 0) $, and $ (0, \infty) $. Pick any test point within each interval.

Q: What if I get confused about the signs?

Remember: negative × negative = positive and negative × positive = negative. In interval $ (-\infty, -2) $, both factors $ 3x $ and $ (x + 2) $ are negative, so the product is positive!

Q: Why don't we include the boundary points x = 0 and x = -2?

The inequality is $ f(x) > 0 $ (strictly greater than), not $ f(x) ≥ 0 $. At $ x = 0 $ and $ x = -2 $, the function equals exactly zero, not greater than zero.

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Look at the following function:

$y=3x^2+6x$

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

$f(x) > 0$

2

Step-by-step solution

To solve the inequality $3x^2 + 6x > 0$ , we first factor the quadratic equation.

Step 1: Factor the quadratic expression:
$3x^2 + 6x = 3x(x + 2)$ .
Step 2: Find the roots by setting the factored expression equal to zero:
- $3x(x + 2) = 0$ gives $x = 0$ and $x = -2$ as roots.
Step 3: Determine the intervals dictated by the roots on a number line:
- The critical points divide the number line into three intervals: $(-\infty, -2)$ , $(-2, 0)$ , and $(0, \infty)$ .
Step 4: Analyze the sign of $3x(x + 2)$ in each interval:

Interval $(-\infty, -2)$ : Pick a test point like $x = -3$ .
$3(-3)((-3) + 2) = 3(-3)(-1) = 9 > 0$ . Hence, $f(x) > 0$ .
Interval $(-2, 0)$ : Pick a test point like $x = -1$ .
$3(-1)((-1) + 2) = 3(-1)(1) = -3 < 0$ . Hence, $f(x) < 0$ .
Interval $(0, \infty)$ : Pick a test point like $x = 1$ .
$3(1)((1) + 2) = 3(1)(3) = 9 > 0$ . Hence, $f(x) > 0$ .

Therefore, the solution to the inequality $3x^2 + 6x > 0$ is:
$x > 0$ or $x < -2$ .

The correct choice from the given options is .

Thus, when $x > 0$ or $x < -2$ , the function $f(x) = 3x^2 + 6x$ is greater than zero.

3

Final Answer

$x > 0$ or $x < -2$

Key Points to Remember

Essential concepts to master this topic

Factoring: Factor out common terms to find critical points
Technique: Use test points like $x = -3$ gives $3(-3)(-1) = 9 > 0$
Check: Substitute boundary values to verify they make expression equal zero ✓

Common Mistakes

Avoid these frequent errors

Solving for when the expression equals zero instead of greater than zero
Don't just find the roots $x = 0$ and $x = -2$ and stop there = incomplete solution! Finding roots only shows where the function changes sign. Always test intervals between roots to determine where the inequality is actually satisfied.

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 factor the expression first?

+

Factoring $3x^2 + 6x = 3x(x + 2)$ reveals the critical points where the function equals zero. These points divide the number line into intervals where the sign stays constant!

How do I know which intervals to test?

+

The roots $x = -2$ and $x = 0$ create three intervals: $(-\infty, -2)$ , $(-2, 0)$ , and $(0, \infty)$ . Pick any test point within each interval.

What if I get confused about the signs?

+

Remember: negative × negative = positive and negative × positive = negative. In interval $(-\infty, -2)$ , both factors $3x$ and $(x + 2)$ are negative, so the product is positive!

Why don't we include the boundary points x = 0 and x = -2?

+

The inequality is $f(x) > 0$ (strictly greater than), not $f(x) ≥ 0$ . At $x = 0$ and $x = -2$ , the function equals exactly zero, not greater than zero.

How can I double-check my final answer?

+

Pick a value from your solution set and substitute it back! For example, try $x = 1$ : $3(1)^2 + 6(1) = 3 + 6 = 9 > 0$ ✓

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Solve the Quadratic Inequality: 3x² + 6x > 0

Quadratic Inequalities with Sign Analysis

❤️ Continue Your Math Journey!

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I need to factor the expression first?

How do I know which intervals to test?

What if I get confused about the signs?

Why don't we include the boundary points x = 0 and x = -2?

How can I double-check my final answer?

🌟 Unlock Your Math Potential

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