To solve the inequality [tex]\(2x^2 + 7x + 2 \geq 0\)[/tex], we need to follow these steps:
1. Identify the quadratic function and find its discriminant:
The given quadratic inequality is [tex]\(2x^2 + 7x + 2 \geq 0\)[/tex].
The standard form of a quadratic equation is [tex]\(ax^2 + bx + c\)[/tex], where [tex]\(a = 2\)[/tex], [tex]\(b = 7\)[/tex], and [tex]\(c = 2\)[/tex].
2. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by [tex]\(\Delta = b^2 - 4ac\)[/tex].
So, [tex]\(\Delta = 7^2 - 4 \cdot 2 \cdot 2 = 49 - 16 = 33\)[/tex].
3. Find the roots of the quadratic equation:
The roots of the quadratic equation are given by:
[tex]\[
x_{1,2} = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substituting the values, we get:
[tex]\[
x_1 = \frac{-7 - \sqrt{33}}{4}
\][/tex]
[tex]\[
x_2 = \frac{-7 + \sqrt{33}}{4}
\][/tex]
4. Determine the intervals where the inequality holds:
Since the coefficient of [tex]\(x^2\)[/tex] (which is 2) is positive, the parabola opens upwards. Therefore, the quadratic expression [tex]\(2x^2 + 7x + 2\)[/tex] will be greater than or equal to zero outside the interval [tex]\([x_1, x_2]\)[/tex].
5. Write down the solution to the inequality:
Based on the interval analysis, the quadratic inequality [tex]\(2x^2 + 7x + 2 \geq 0\)[/tex] is satisfied for [tex]\(x \leq \frac{-7 - \sqrt{33}}{4}\)[/tex] or [tex]\(x \geq \frac{-7 + \sqrt{33}}{4}\)[/tex].
Among the given choices, this corresponds to option C:
[tex]\[
\boxed{x \leq \frac{-7 - \sqrt{33}}{4} \text{ or } x \geq \frac{-7 + \sqrt{33}}{4}}
\][/tex]
