Final answer:
The quadratic inequality x²-3x-10>0 is solved by factoring the expression and testing intervals determined by its roots (-2 and 5). The solution in interval notation is (-∞, -2) ∪ (5, ∞).
Step-by-step explanation:
To solve the quadratic inequality x²-3x-10>0, first, we should factor the quadratic expression. The factors of -10 that add up to -3 are -5 and 2. Thus, we can rewrite the inequality as:
(x - 5)(x + 2) > 0
The roots of the inequality are x = 5 and x = -2. To determine the intervals where the inequality is positive, we test values from the intervals created by the roots:
- Test x = -3, which lies in the interval (-∞, -2): (-3 - 5)(-3 + 2) = (8)(-1) < 0
- Test x = 0, which lies in the interval (-2, 5): (0 - 5)(0 + 2) = (-5)(2) < 0
- Test x = 6, which lies in the interval (5, ∞): (6 - 5)(6 + 2) = (1)(8) > 0
The inequality is satisfied for x in the intervals (-∞, -2) and (5, ∞). Thus, the solution set in interval notation is:
(-∞, -2) ∪ (5, ∞)