Certainly! Let's solve the inequality [tex]\( -x^2 + 2x + 8 < 0 \)[/tex].
1. Rearrange the Inequality: The given inequality is already in a standard quadratic form:
[tex]\[
-x^2 + 2x + 8 < 0
\][/tex]
2. Identify the Quadratic Function: This can be rewritten as:
[tex]\[
f(x) = -x^2 + 2x + 8
\][/tex]
We need to find where [tex]\( f(x) \)[/tex] is less than zero.
3. Find the Roots of the Quadratic: First, solve the equation [tex]\(-x^2 + 2x + 8 = 0\)[/tex] to find the boundary points. However, we need to remember what we are looking for is the intervals where the quadratic expression is negative (i.e., below the x-axis).
4. Analyze the Sign of the Quadratic: Let's analyze where [tex]\( f(x) = -x^2 + 2x + 8 \)[/tex] is negative by considering the intervals around the roots of the equation.
5. Solution: From calculations and analyzing the behavior of the quadratic function, we find that the function is less than zero in the intervals:
[tex]\[
(-\infty, -2) \cup (4, \infty)
\][/tex]
So, the solution to the inequality [tex]\( -x^2 + 2x + 8 < 0 \)[/tex] is:
[tex]\[
(-\infty, -2) \cup (4, \infty)
\][/tex]
