The smallest negative integer satisfying both the quadratic inequalities x 2 < 4 x + 77 and x 2 > 4 is

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The smallest negative integer satisfying both the quadratic inequalities Given, x 2 < 4 x + 77 and x 2 > 4 x 2 - 4 x < 77 Adding 4 on both sides x 2 - 4 x + 4 < 77 + 4 x - 2 2 < 81 - 9 < x - 2 < 9 - 7 < x < 11 . . . . . . . . . . . . . . . . . . . . . . . . i Now, x 2 > 4 x 2 - 4 > 0 x > 2   or x < - 2 . . . . . . . . . . . . . . . . . . i i The solution set satisfying both inequalities is the common part displayed in the diagram, namely, - 7 , - 2 ∪ 2