Solving x²-6x+8 > 0: Analyzing Quadratic Inequality with Graphs

To solve the inequality f(x) > 0 , we first need to find the roots of the equation $f(x) = x^2 - 6x + 8$.

1. Find the roots of the quadratic equation:
The quadratic is $x^2 - 6x + 8$. This can be factored into:
$f(x) = (x - 2)(x - 4) = 0$.

2. Calculate the roots:
Setting each factor equal to zero gives the roots $x = 2$ and $x = 4$.

3. Determine the intervals defined by these roots:
The roots divide the x-axis into three intervals: $(-\infty, 2)$, $(2, 4)$, and $(4, \infty)$.

4. Test points in each interval to decide positivity:
- For x < 2 , select $x = 1$: f(1) = 1^2 - 6(1) + 8 = 3 > 0 . Thus, f(x) > 0 in $(-\infty, 2)$.
- For 2 < x < 4 , select $x = 3$: f(3) = 3^2 - 6(3) + 8 = -1 < 0 . Thus, f(x) < 0 in $(2, 4)$.
- For x > 4 , select $x = 5$: f(5) = 5^2 - 6(5) + 8 = 3 > 0 . Thus, f(x) > 0 in $(4, \infty)$.

Therefore, the solution to f(x) > 0 is when x < 2 or x > 4 .

The final solution is: x < 2, 4 < x .