To find the endpoints of the solution intervals for the quadratic inequality [tex]\( x^2 + 6x > 16 \)[/tex], we need to treat this as a quadratic equation initially to find the critical points where the quadratic expression equals 16. Follow these steps:
1. Rewrite the Inequality:
Rewrite the inequality as an equality:
[tex]\[
x^2 + 6x = 16
\][/tex]
2. Transform to Standard Form:
Move 16 to the left side to get a standard quadratic equation:
[tex]\[
x^2 + 6x - 16 = 0
\][/tex]
3. Solve the Quadratic Equation:
Use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] where [tex]\( a = 1 \)[/tex], [tex]\( b = 6 \)[/tex], and [tex]\( c = -16 \)[/tex]:
[tex]\[
x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot (-16)}}{2 \cdot 1}
\][/tex]
Simplify inside the square root:
[tex]\[
x = \frac{-6 \pm \sqrt{36 + 64}}{2}
\][/tex]
[tex]\[
x = \frac{-6 \pm \sqrt{100}}{2}
\][/tex]
4. Find the Roots:
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-6 \pm 10}{2}
\][/tex]
This gives two solutions:
[tex]\[
x = \frac{-6 + 10}{2} = \frac{4}{2} = 2
\][/tex]
[tex]\[
x = \frac{-6 - 10}{2} = \frac{-16}{2} = -8
\][/tex]
5. Check the Intervals:
With the roots [tex]\( x = 2 \)[/tex] and [tex]\( x = -8 \)[/tex], these are the critical points where [tex]\( x^2 + 6x = 16 \)[/tex]. The inequality [tex]\( x^2 + 6x > 16 \)[/tex] requires you to check the intervals determined by these points:
- For [tex]\( x < -8 \)[/tex]
- For [tex]\( -8 < x < 2 \)[/tex]
- For [tex]\( x > 2 \)[/tex]
Since [tex]\( x^2 + 6x \)[/tex] is a parabola opening upwards, [tex]\( x^2 + 6x - 16 \)[/tex] will be negative between the roots and positive outside them. Hence, the solution to the inequality [tex]\( x^2 + 6x > 16 \)[/tex] are the intervals where the expression is positive:
[tex]\[
x \in (-\infty, -8) \cup (2, \infty)
\][/tex]
6. Identify the Endpoints:
The endpoints of the solution intervals are [tex]\( x = -8 \)[/tex] and [tex]\( x = 2 \)[/tex].
So, the correct answer is [tex]\( x = -8 \)[/tex] and [tex]\( x = 2 \)[/tex].
