Which of the options is the solution to the following inequality?
2
𝑥
2
+
𝑥
≥
5

Question

BlackTom AI · Accepted Answer

A quadratic inequality is treated by bringing all terms to one side and analyzing the sign of the quadratic. Move 5 to the left to get
2x^2 + x − 5 ≥ 0.
1) Relevant concepts: roots of the quadratic (where it equals zero), discriminant to find exact roots, and the parabola's opening direction (coefficient a) to determine sign regions.

2) Find the roots using the quadratic formula x = [−b ± √(b^2 − 4ac)]/(2a) with a = 2, b = 1, c = −5.
Compute the discriminant:
Δ = b^2 − 4ac = 1^2 − 4(2)(−5) = 1 + 40 = 41.
Thus the roots are
x = [−1 ± √41]/(4).

3) Intermediate numeric check (optional): √41 ≈ 6.403, so roots ≈ (−1 − 6.403)/4 ≈ −1.85075 and (−1 + 6.403)/4 ≈ 1.35075.

4) Because a = 2 > 0 the parabola opens upward. Therefore the quadratic 2x^2 + x − 5 is ≥ 0 outside (to the left of the smaller root and to the right of the larger root), and ≤ 0 between the roots. The inequality ≥ 0 includes equality, so the roots themselves are included.

5) Solution set in interval notation:
(−∞, (−1 − √41)/4 ] ∪ [ (−1 + √41)/4, ∞).

6) Why other common alternatives would be incorrect: the interval between the roots would correspond to ≤ 0 (not ≥ 0); omitting the endpoints would be wrong because ≥ includes equality at the roots; sign errors (e.g., flipping inequality) or miscomputing the discriminant would yield incorrect bounds.

Conclusion: The correct solution is (−∞, (−1 − √41)/4 ] ∪ [ (−1 + √41)/4, ∞), which matches the provided answer.

Which of the options is the solution to the following inequality? 2 𝑥 2 + 𝑥 ≥ 5单项选择题

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