Solve the Quadratic Inequality: x² - 8x + 12 < 0

To solve the quadratic inequality $x^2 - 8x + 12 < 0$, we follow these steps:

• Step 1: Factor the quadratic expression. The expression $x^2 - 8x + 12$ can be factored as $(x - 2)(x - 6)$.
• Step 2: Find the roots of the resulting equation. Set $(x - 2)(x - 6) = 0$ to find the roots. Solving gives $x = 2$ and $x = 6$.
• Step 3: Determine the intervals for testing. The roots divide the number line into three intervals: $x < 2$, $2 < x < 6$, and $x > 6$.
• Step 4: Test each interval to find where the inequality holds.
• For $x < 2$, choose a test point such as $x = 0$. Then $(0 - 2)(0 - 6) = ( -2)( -6) = 12$, which is greater than 0.
• For $2 < x < 6$, choose a test point such as $x = 4$. Then $(4 - 2)(4 - 6) = (2)(-2) = -4$, which is less than 0.
• For $x > 6$, choose a test point such as $x = 7$. Then $(7 - 2)(7 - 6) = (5)(1) = 5$, which is greater than 0.
• Step 5: Conclude the solution. The expression $(x - 2)(x - 6)$ is negative in the interval $2 < x < 6$.

Therefore, the solution to the inequality is $2 < x < 6$.