Identify Values of X for Positivity in the Quadratic y = 3x² - 27

To solve this problem, we'll follow these steps:

• Step 1: Solve the quadratic equation to find the roots.
• Step 2: Determine intervals based on these roots.
• Step 3: Test the sign of the function on each interval.

Now, let's work through each step:

Step 1: The given function is $y = 3x^2 - 27$. To find where this function equals zero, solve the equation $3x^2 - 27 = 0$.

Factor the equation:
$3x^2 - 27 = 3(x^2 - 9) = 3(x - 3)(x + 3) = 0$. The solutions are $x = 3$ and $x = -3$.

Step 2: The critical points from step 1 divide the number line into three intervals: $x < -3$, $-3 < x < 3$, and $x > 3$.

Step 3: Test each interval:

• For $x < -3$: Choose a test value like $x = -4$. Plug it into the inequality $3x^2 - 27 > 0$:
  $y = 3(-4)^2 - 27 = 3(16) - 27 = 48 - 27 = 21 > 0$. The function is positive.
• For $-3 < x < 3$: Choose a test value like $x = 0$. Plug it into the inequality:
  $y = 3(0)^2 - 27 = -27 < 0$. The function is negative.
• For $x > 3$: Choose a test value like $x = 4$. Plug it into the inequality:
  $y = 3(4)^2 - 27 = 48 - 27 = 21 > 0$. The function is positive.

We conclude that the function is positive for $x > 3$ or $x < -3$.

Therefore, the solution to the problem is $x > 3$ or $x < -3$.