Solve the Quadratic Inequality: When Is x² - 36 Positive?

To solve this problem, we will perform the following steps:

• Step 1: Factor the quadratic expression.
• Step 2: Determine the roots of the function.
• Step 3: Analyze intervals around the roots to find where the function is positive.

Step 1: The quadratic function given is $f(x) = x^2 - 36$. We factor it as follows:

$f(x) = (x - 6)(x + 6)$

Step 2: The roots of the quadratic are found by setting the factored form to zero:

$x - 6 = 0$ or $x + 6 = 0$, giving roots $x = 6$ and $x = -6$.

Step 3: We now analyze the function between the roots to determine the intervals where $f(x) > 0$.

• Interval 1: $x < -6$
  In this region, choose a test point, such as $x = -7$. Substituting into the factored form gives:
  $f(-7) = (-7 - 6)(-7 + 6) = ( -13)\cdot(-1) = 13$, which is positive.
• Interval 2: $-6 < x < 6$
  Choose a test point, such as $x = 0$. Substituting into the factored form gives:
  $f(0) = (0 - 6)(0 + 6) = (-6)\cdot(6) = -36$, which is negative.
• Interval 3: $x > 6$
  Choose a test point, such as $x = 7$. Substituting into the factored form gives:
  $f(7) = (7 - 6)(7 + 6) = (1)\cdot(13) = 13$, which is positive.

The function $f(x) = x^2 - 36$ is positive in the intervals $x > 6$ and $x < -6$.

Therefore, the solution to the problem is that the function is positive for $x > 6$ or $x < -6$.

Therefore, the answer is: $x > 6$ or $x < -6$.