a)
[tex]4x^{2}+3x-1 > 0\\\\a = 4, \ b = 3, \ c = -1\\\\\Delta = b^{2}-4ac = 3^{2}-4\cdot4\cdot(-1) = 9+16 = 25\\\\\sqrt{\Delta} = \sqrt{25} = 5\\\\x_1 = \frac{-b-\sqrt{\Delta}}{2a} = \frac{-3-5}{2\cdot4} = \frac{-8}{8} = -1\\\\x_2 = \frac{-b+\sqrt{\Delta}}{2a} = \frac{-3+5}{8} = \frac{2}{8}=\frac{1}{4}\\\\a > 0, \ ramiona \ paraboli \ skierowane \ do \ gory[/tex]
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---------o--------0--o------------->
-1 - ¹/₄
[tex]x \in (-\infty;-1) \ \cup \ (\frac{1}{4};+\infty)[/tex]
b)
[tex]-x^{2}-7x-10 \leq 0\\\\a = -1, \ b = -7, \ c = -10\\\\\Delta = b^{2}-4ac = (-7)^{2}-4\cdot(-1)\cdot(-10) = 49-40 = 9\\\\\sqrt{\Delta}=\sqrt{9} = 3\\\\x_1 = \frac{7+3}{2\cdot(-1)} = \frac{10}{-2} = -5\\\\x_2 = \frac{7-3}{-2} = \frac{4}{-2} = -2\\\\a < 0, \ ramiona \ paraboli \ skierowane \ do \ dolu[/tex]
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----------.------------------.---------------> x
- - - -5 -2 - - -
[tex]x \in (-\infty;-5\rangle \ \cup \ \langle-2;+\infty)[/tex]
c)
[tex]2x^{2}-3x+1 \geq 0\\\\a = 2, \ b = -3, \ c = 1\\\\\Delta = b^{2}-4ac = (-3)^{2}-4\cdot2\cdot1 = 9-8 = 1\\\\\sqrt{\Delta} = \sqrt{1} = 1\\\\x_1 = \frac{3-1}{2\cdot2} = \frac{2}{4} = \frac{1}{2}\\\\x_2 = \frac{3+1}{4} = \frac{4}{4} = 1\\\\a > 0, \ ramiona \ paraboli \ skierowane \ do \ gory[/tex]
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----------------0----.--------.-------------------> x
¹/₂ - 1
[tex]x \in (-\infty;\frac{1}{2}\rangle \ \cup \ \langle1;+\infty)[/tex]
