[latex]a)2(x+1)^2=5(4-x) \2(x^2+2x+1)=20-5x \2x^2+4x+2=20-5x \2x^2+9x-18=0 \Delta=81+144=225 Rightarrow sqrt{Delta}=15 \x_1= frac{-9-15}{4}=-6 \x_2= frac{-9+15}{4}= frac{3}{2} [/latex] [latex]b)x^2+ sqrt{2}x-4=0 \Delta=2+16=18 Rightarrow sqrt{Delta}=3 sqrt{2} \x_1= frac{- sqrt{2}-3 sqrt{2} }{2} =-2 sqrt{2} \x_2= frac{- sqrt{2}+3 sqrt{2} }{2} = sqrt{2} [/latex] [latex]c) sqrt{3} x^2-9x+6sqrt{3}=0 \Delta=81-72=9 Rightarrow sqrt{Delta}=3 \x_1= frac{9-3}{2 sqrt{3} } = frac{6sqrt{3}}{2*3}=sqrt{3} \x_2= frac{9+3}{2sqrt{3}}= frac{12sqrt{3}}{2*3} =2sqrt{3} [/latex] ---------------------------- [latex]a)x^2-9>0 \(x-3)(x+3)>0 \x-3=0 ; vee ;x+3=0 \x=3 ; vee ; x=-3 \\underline{+++++}(-3)underline{-----}(3)underline{++++++} \\x in (-infty;-3) cup (3;+infty)[/latex] [latex]b)6x^2+3x geq 0 \3x(2x+1) geq 0 \3x = 0 ; vee ; 2x+1 = 0 \x =0 ; vee ;x = - frac{1}{2} \\underline{+++++}langle - frac{1}{2} angle underline{-----}langle0 angleunderline{+++++} \\x in (-infty;- frac{1}{2} angle cup langle 0;+infty)[/latex] [latex]c)x^2+16 geq 8x \x^2-8x+16 geq 0 \(x-4)^2 geq 0 \x in mathbf{R}[/latex] [latex]d)2x^2-2+3x leq 0 \2x^2+3x-2 leq 0 \Delta=9+16=25 Rightarrow sqrt{Delta}=5 \x_1= frac{-3-5}{4}=-2 \x_2= frac{-3+5}{4}= frac{1}{2} \x in langle -2; frac{1}{2} angle [/latex] [latex]e)2x^2+5x-3>0 \Delta=25+24=49 Rightarrow sqrt{Delta}=7 \x_1= frac{-5-7}{4}=-3 \x_2= frac{-5+7}{4}= frac{1}{2} \x in (-infty;-3) cup ( frac{1}{2};+infty) [/latex] [latex]f)-3x^2-2x+1 leq 0 \Delta=4+12=16 Rightarrow sqrt{Delta}=4 \x_1= frac{2-4}{-6}= frac{1}{3} \x_2= frac{2+4}{-6}=-1 \x in (-infty ; -1 angle cup langle frac{1}{3} ,+infty) [/latex]
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