Quadratic Equations

To find the common solution set for the inequalities x2−4x≤12 and x2−2x≥15, we'll begin by rewriting them in a standard form:
‌x2−4x−12≤0
‌x2−2x−15≥0
Next, we factor each equation:
‌(x−6)(x+2)≤0
‌(x−5)(x+3)≥0
Now, we determine the solution sets for each inequality:
For (x−6)(x+2)≤0, the interval is x∈[−2,6].
For (x−5)(x+3)≥0, the intervals are x∈(−∞,−3]∪[5,∞).
To find the common solution set, we identify where these intervals overlap:
The intersection of [−2,6] and (−∞,−3]∪[5,∞) is x∈[5,6].
Thus, the common solution set for the given inequalities is x∈[5,6].