Solve (x+6)(x-3) > 0: Finding Positive Values of a Quadratic Function

Look at the following function:

$y=\left(x+6\right)\left(x-3\right)$

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

$f(x) > 0$

To determine for which values of $x$ the function $y = (x + 6)(x - 3)$ is positive, let's work through the steps:

• Step 1: Identify the roots of the quadratic equation. Set the equation equal to zero: $(x + 6)(x - 3) = 0$.
  The roots are $x = -6$ and $x = 3$.
• Step 2: Analyze the sign of the function in the intervals around the roots:
  - Interval 1: $x < -6$
  - Interval 2: $-6 < x < 3$
  - Interval 3: $x > 3$
• Step 3: Test the sign of the quadratic expression in each interval:
• In Interval 1 ($x < -6$): Choose $x = -7$:
  $(x + 6) = -7 + 6 = -1$; $(x - 3) = -7 - 3 = -10$
  The product $(x + 6)(x - 3) = (-1)(-10) = 10$, which is positive.
• In Interval 2 ($-6 < x < 3$): Choose $x = 0$:
  $(x + 6) = 0 + 6 = 6$; $(x - 3) = 0 - 3 = -3$
  The product $(x + 6)(x - 3) = (6)(-3) = -18$, which is negative.
• In Interval 3 ($x > 3$): Choose $x = 4$:
  $(x + 6) = 4 + 6 = 10$; $(x - 3) = 4 - 3 = 1$
  The product $(x + 6)(x - 3) = (10)(1) = 10$, which is positive.

Step 4: Conclusion:
The inequality $f(x) > 0$ holds for $x$ in Interval 1 ($x < -6$) and Interval 3 ($x > 3$).
Therefore, the values of $x$ that satisfy $f(x) > 0$ are $x < -6$ or $x > 3$.

The solution to the problem is $x > 3$ or $x < -6$.

Thus, the correct choice among the provided options is Choice 4.