Solve the Quadratic Inequality: Finding When x²+6x>0

To solve the inequality $x^2 + 6x > 0$, follow these steps:

• Step 1: Write the inequality in factored form.
  Express $x^2 + 6x$ as $x(x + 6)$.
• Step 2: Identify the roots of the equation $x(x + 6) = 0$.
  The roots are $x = 0$ and $x = -6$.
• Step 3: Determine the sign of $x(x + 6)$ in each interval divided by the roots.
• Step 4: Test three intervals: $x < -6$, $-6 < x < 0$, and $x > 0$.

For $x < -6$:
Pick a value such as $x = -7$. Substituting, $x(x + 6) = (-7)(-7 + 6) = (-7)(-1) = 7 > 0$.
Thus, $x^2 + 6x > 0$ for $x < -6$.

For $-6 < x < 0$:
Pick a value such as $x = -3$. Substituting, $x(x + 6) = (-3)(-3 + 6) = (-3)(3) = -9 < 0$.
Thus, $x^2 + 6x < 0$ for $-6 < x < 0$.

For $x > 0$:
Pick a value such as $x = 1$. Substituting, $x(x + 6) = (1)(1 + 6) = 1 \times 7 = 7 > 0$.
Thus, $x^2 + 6x > 0$ for $x > 0$.

Therefore, the solution to the inequality is $x < -6$ or $x > 0$.

Thus, the correct answer is $x < -6, 0 < x$.