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Rozwiazania sa podkreslone
[tex]\bold{1. }\\\frac{3x-1}{7x+1}=\frac25\\5(3x-1)=2(7x+1)\\15x-5=14x+2\\15x-14x=2+5\\\underline{x=7}\\\\\bold{2. }\\(x-2)(x+3) < 0\\a > 0 - \text{ramiona skierowane w gore}\\x_1=2\\x_2=-3\\\underline{x\in(-3; 2)}\\\\\bold{3.}\\x^2-x-2 \leq 0\\a > 0 - \text{ramiona skierowane w gore}\\\Delta=(-1)^2-4*1*(-2)=1+8=9\\\sqrt{\Delta}=3\\x_1=\frac{1-3}{2}=\frac{-2}2=-1\\x_2=\frac{1+3}2=\frac42=2\\\underline{x\in < -1; 2 > }[/tex]
[tex]\bold{4. }\\x^3-7x^2-4x+28=0\\x^2(x-7)-4(x-7)=0\\(x^2-4)(x-7)=0\\(x-2)(x+2)(x-7)=0\\x-2=0 \text{ v } x+2=0 \text{ v } x-7=0\\\underline{x=2 \text{ v } x=-2 \text{ v } x=7}\\\\\bold{5.}\\x^2-14x+24 > 0\\a > 0 - \text{ramiona skierowane w gore}\\\Delta=(-14)^2-4*1*24=196-96=100\\\sqrt{\Delta}=10\\x_1=\frac{14-10}2=\frac42=2\\x_2=\frac{14+10}2=\frac{24}2=12\\\underline{x\in(-\infty; 2)U(12; \infty)}[/tex]
[tex]\bold{6. }\\x^3-3x^2+2x-6=0\\x^2(x-3)+2(x-3)=0\\(x^2+2)(x-3)=0\\x^2+2=0 \text{ v } x-3=0\\\underline{x=3}[/tex]
[tex]\bold{7.}\\3x^2-10x+3 \leq0\\a > 0 - \text{ramiona skierowane w gore}\\\Delta=(-10)^2-4*3*3=100-36=64\\\sqrt{\Delta}=8\\x_1=\frac{10-8}6=\frac26=\frac13\\x_2=\frac{10+8}6=\frac{18}6=3\\\underline{x\in < \frac13; 3 > }[/tex]
[tex]\bold{8.}\\\frac{x^2+25}{x-5}=0\\x-5\neq 0 /+5\\x\neq 5\\D\in R / \{5\}\\\\x^2+25=0\\\Delta=0^2-4*1*25=-100 - \text{Delta ujemna, \underline{brak rozwiazan}}[/tex]
[tex]\bold{9.}\\-2x^2+2x+24 \geq0\\a < 0 - \text{ramiona skierowane w dol}\\\Delta=2^2-4*(-2)*24=4+192=196\\\sqrt{\Delta}=14\\x_1=\frac{-2-14}{-4}=\frac{-16}{-4}=4\\x_2=\frac{-2+14}{-4}=\frac{12}{-4}=-3\\\underline{x\in < -3; 4 > }[/tex]
[tex]\bold{10.}\\x^2-3x+2 < 0\\a > 0 - \text{ ramiona skierowane w gore}\\\Delta=(-3)^2-4*1*2=9-8=1\\\sqrt{\Delta}=1\\x_1=\frac{3-1}2=\frac22=1\\x_2=\frac{3+1}2=\frac42=2\\\underline{x\in(1, 2)}[/tex]
[tex]\bold{11. }\\x^2+8x+15 > 0\\a > 0 - \text{ramiona skierowane w gore}\\\Delta=8^2-4*1*15=64-60=4\\\sqrt{\Delta}=2\\x_1=\frac{-8-2}2=\frac{-10}2=-5\\x_2=\frac{-8+2}2=\frac{-6}2=-3\\\underline{x\in(-\infty; -5)U(-3; \infty)}\\\\\bold{12. }\\(x+5)(x-3)(x^2+1)=0\\x+5=0 \text{ v } x-3=0 \text{ v } x^2+1=0\\\underline{x=-5 \text{ v } x=3}[/tex]
[tex]\bold{13.}\\x^2-3x-10 < 0\\a > 0\\\Delta=(-3)^2-4*1*(-10)=9+40=49\\\sqrt{\Delta}=7\\x_1=\frac{3-7}2=\frac{-4}2=-2\\x_2=\frac{3+7}2=\frac{10}2=5\\\underline{x\in(-2; 5)}\\\\\bold{14. }\\x(x+6) < 0\\x^2+6x < 0\\a > 0\\\Delta=6^2-4*1*0=36\\\sqrt{\Delta}=6\\x_1=\frac{-6-6}{2}=\frac{-12}2=-6\\x_2=\frac{-6+6}2=0\\\underline{x\in(-6; 0)}\\\\[/tex]
[tex]\bold{15. }\\x^2 \geq 8x-7\\x^2-8x+7\geq 0\\a > 0\\\Delta=(-8)^2-4*1*7=64-28=36\\\sqrt{\Delta}=6\\x_1=\frac{8-6}2=\frac22=1\\x_2=\frac{8+6}2=\frac{14}2=7\\\underline{x\in(-\infty; 1 > U < 7; \infty)}[/tex]
[tex]\bold{16.}\\x^3-6x^2-9x+54=0\\x^2(x-6)-9(x-6)=0\\(x^2-9)(x-6)=0\\(x-3)(x+3)(x-6)=0\\x-3=0 \text{ v } x+3=0 \text{ v } x-6=0\\\underline{x=3 \text{ v } x=-3 \text{ v } x=6}[/tex]
[tex]\bold{17.}\\x^3+2x^2-8x-16=0\\x^2(x+2)-8(x+2)=0\\(x^2-8)(x+2)=0\\(x-2\sqrt2)(x+2\sqrt2)(x+2)=0\\x-2\sqrt2=0 \text{ v } x+2\sqrt2=0 \text{ v } x+2=0\\\underline{x=2\sqrt2 \text{ v } x=-2\sqrt2 \text{ v } x=-2}[/tex]
[tex]\bold{18.}\\2x^2-7x+5 \geq 0\\a > 0\\\Delta=(-7)^2-4*2*5=49-40=9\\\sqrt{\Delta}=3\\x_1=\frac{7-3}4=\frac{4}4=1\\x_2=\frac{7+3}4=\frac{10}4=\frac52\\\underline{x\in (-\infty; 1 > U < \frac52; \infty)}[/tex]
[tex]\bold{19.}\\3x^3-4x^2-3x+4=0\\x^2(3x-4)-1(3x-4)=0\\(x^2-1)(3x-4)=0\\(x-1)(x+1)(3x-4)=0\\x-1=0 \text{ v } x+1=0 \text{ v } 3x-4=0\\\underline{x=1 \text{ v } x=-1 \text{ v } x=\frac43}[/tex]
[tex]\bold{20.}\\2(3-x) > x\\6-2x > x\\6 > x+2x\\6 > 3x /:3\\2 > x\\x < 2\\\underline{x\in(-\infty; 2)}\\\\\bold{21.}\\3x-x^2 \geq0\\-x^2+3x\geq0\\a < 0\\\Delta=3^2-4*(-1)*0=9\\\sqrt{\Delta}=3\\x_1=\frac{-3-3}{-2}=\frac{-6}{-2}=3\\x_2=\frac{-3+3}{-2}=0\\\underline{x\in < 0; 3 > }[/tex]
[tex]\bold{22.}\\x^3-6x^2-12x+72=0\\x^2(x-6)-12(x-6)=0\\(x^2-12)(x-6)=0\\(x-2\sqrt3)(x+2\sqrt3)(x-6)=0\\x-2\sqrt3=0 \text{ v } x+2\sqrt3=0 \text{ v } x-6=0\\\underline{x=2\sqrt3 \text{ v } x=2\sqrt3 \text{ v } x=6}[/tex]
[tex]\bold{23.}\\(2x-3)(3-x)\geq0\\2x-3=0 \text{ v } 3-x=0\\x_1=\frac32 \text{ v } x_2=3\\a < 0\\\underline{x\in < \frac32; 3 > }\\\\\bold{24}\\\frac{x-5}{7-x}=\frac13\\3(x-5)=7-x\\3x-15=7-x\\3x+x=7+15\\4x=22 /:4\\x=\frac{22}4=\frac{11}2=5\frac12[/tex]
[tex]\bold{25.}\\-x^2-5x+14 < 0\\a < 0\\\Delta=(-5)^2-4*(-1)*14=25+56=81\\\sqrt{\Delta}=9\\x_1=\frac{5-9}{-2}=\frac{-4}{-2}=2\\x_2=\frac{5+9}{-2}=\frac{14}{-2}=-7\\\underline{x\in (-\infty; -7)U(2; \infty)}\\\\\bold{26.}\\\frac{x-1}{x+1}=x-1 /*(x+1)\\x-1=(x-1)(x+1)\\x-1=x^2-1\\-x^2+x-1+1=0\\-x^2+x=0\\a < 0\\\Delta=1^2-4*(-1)*0=1\\\sqrt{\Delta}=1\\\underline{x_1=\frac{-1-1}{-2}=\frac{-2}{-2}=1}\\\underline{x_2=\frac{-1+1}{-2}=0}\\[/tex]
[tex]\bold{27.}\\2x^2-4x > (x+3)(x-2)\\2x^2-4x > x^2+x-6\\2x^2-x^2-4x-x+6 > 0\\x^2-5x+6 > 0\\a > 0\\\Delta=(-5)^2-4*1*6=25-24=1\\\sqrt{\Delta}=1\\x_1=\frac{5-1}2=\frac42=2\\x_2=\frac{5+1}2=\frac62=3\\\underline{x\in(-\infty; 2)U(3; \infty)}[/tex]
[tex]\bold{28.}\\\frac{2x-4}{3-x}=\frac43\\3(2x-4)=4(3-x)\\6x-12=12-4x\\6x+4x=12+12\\10x=24 /:10\\\underline{x=2.4}[/tex]
[tex]\bold{29.}\\2x^2-4x\geq x-2\\2x^2-4x-x+2\geq0\\2x^2-5x+2\geq0\\a > 0\\\Delta=(-5)^2-4*2*2=25-16=9\\\sqrt{\Delta}=3\\x_1=\frac{5-3}{4}=\frac{2}4=\frac12\\x_2=\frac{5+3}4=\frac84=2\\\underline{x\in (-\infty; \frac12 > U < 2; \infty)}[/tex]
[tex]\bold{30.}\\4x^3+4x^2-x-1=0\\4x^2(x+1)-1(x+1)=0\\(4x^2-1)(x+1)=0\\4x^2-1=0 \text{ v } x+1=0\\x^2=\frac14 \text{ v } x=-1\\\underline{x=\frac12 \text{ v } x=-\frac12 \text{ v } x=-1 }[/tex]
[tex]\bold{31.}\\x^2(x+5)(2x-3)(x^2-7)=0\\x^2=0 \text{ v } x+5=0 \text{ v } 2x-3=0 \text{ v } x^2-7=0\\x=0 \text{ v } x=-5 \text{ v } 2x=3 \text{ v } x^2=7\\\underline{x=0 \text{ v } x=-5 \text{ v } x=\frac32 \text{ v } x=\sqrt7 \text{ v } x=-\sqrt7}[/tex]
[tex]\bold{32. }\\7x^2-28 \leq 0\\a > 0\\\Delta=0^2-4*7*(-28)=784\\\sqrt{\Delta}=28\\x_1=\frac{-28}{14}=-2\\x_2=\frac{28}{14}=2\\\underline{x\in < -2; 2 > }[/tex]
[tex]\bold{33.}\\x^4-2x^3+27x-54=0\\x^3(x-2)+27(x-2)=0\\(x^3+27)(x-2)=0\\x^3+27=0 \text{ v } x-2=0\\x^3=-27 \text{ v } x=2\\\underline{x=-3 \text{ v } x=2}[/tex]
[tex]\bold{34.}\\20x \geq 4x^2+24\\-4x^2+20x-24 \geq 0\\a < 0\\\Delta=20^2-4*(-4)*(-24)=400-384=16\\\sqrt{\Delta}=4\\x_1=\frac{-20-4}{-8}=\frac{-24}{-8}=3\\x_2=\frac{-20+4}{-8}=\frac{-16}{-8}=2\\\underline{x\in < 2; 3 > }[/tex]
[tex]\bold{35.}\\2x^2-4x > 3x^2-6x\\2x^2-3x^2-4x+6x > 0\\-x^2+2x > 0\\a < 0\\\Delta=2^2-4*(-1)*0=4\\\sqrt{\Delta}=2\\x_1=\frac{-2-2}{-2}=\frac{-4}{-2}=2\\x_2=\frac{-2+2}{-2}=0\\\underline{x\in(0; 2)}[/tex]