a)
[tex]x^{2}-3x-4 > 0\\\\a = 1, \ b = -3, \ c = -4\\\\M. \ zerowe:\\\\\Delta = (-3)^{2}-4\cdot1\cdot(-4) = 9 + 16 = 25\\\\\sqrt{\Delta} = \sqrt{25} = 5\\\\x_1 = \frac{-b-\sqrt{\Delta}}{2a} = \frac{-(-3)-5}{2\cdot1} = \frac{3-5}{2} = \frac{-2}{2} = -1\\\\x_2 = \frac{-b+\sqrt{\Delta}}{2a} = \frac{-(-3)+5}{2} = \frac{8}{2} = 4[/tex]
a > 0, to parabola zwrócona jest ramionami do góry, wówczas;
x ∈ (-∞;-1) ∪ (4;+∞)
b)
[tex]-x^{2}+8x-16 < 0\\\\a = -1, \ b = 8, \ c = -16\\\\\Delta = b^{2}-4ac = 8^{2}-4\cdot(-1)\cdot(-16) =64-64 = 0\\\\x = \frac{-b}{2a} = \frac{-8}{-2} = 4[/tex]
a < 0, to parabola zwrócona jest ramionami do dołu, wówczas:
x ∈ (-∞; 4) ∪ (4; +∞)
c)
[tex]-x^{2}-4x+5 \geq 0\\\\a = -1, \ b = -4, \ c = 5\\\\\Delta =b^{2}-4ac = (-4)^{2}-4\cdot(-1)\cdot5 = 16+20 = 36\\\\\sqrt{\Delta} = \sqrt{36} = 6\\\\x_1 =\frac{-b+\sqrt{\Delta}}{2a}= \frac{4+6}{-2} = \frac{10}{-2} = -5\\\\x_2=\frac{-b-\sqrt{\Delta}}{2a} = \frac{4-6}{-2} = \frac{-2}{-2} = 1[/tex]
a < 0, to parabola zwrócona jest ramionami do dołu, wówczas:
x ∈ < -5; 1 >
d)
[tex]x^{2}-10x + 25 > 0\\\\M. \ zerowe\\\\(x-5)^{2} = 0\\\\x-5 = 0\\\\x = 5[/tex]
a > 0, to ramiona paraboli zwrócane są do góry, wówczas:
x ∈ R \ {5}
