Rozwiąż równanie. PROSZĘ O SZYBKIE I DOKŁADNE ROZWIĄZANIE DAJĘ NAJ.

Korzystamy ze wzorów skróconego mnożenia: [latex](a-b)^{2}=a^{2}-2ab+b^{2}\(a+b)^{2}=a^{2}+2ab+b^{2}[/latex] d) [latex]9x^{2}-(2x-1)^{2}=0\9x^{2}-(4x^{2}-4x+1)=0\9x^{2}-4x^{2}+4x-1=0\5x^{2}+4x-1=0\Delta=b^{2}-4ac=4^{2}-4*5*(-1)=16+20=36\sqrt{Delta}=6\x_{1}=frac{-b-sqrt{Delta}}{2a}=frac{-4-6}{10}=frac{-10}{10}=-1\\x_{2}=frac{-b+sqrt{Delta}}{2a}=frac{-4+6}{10}=frac{2}{10}=frac{1}{5} \\oxed{xin {-1; frac{1}{5}}}[/latex] e) [latex]4x^{2}=(1-x)^{2}\4x^{2}=1-2x+x^{2}\4x^{2}-x^{2}+2x-1=0\3x^{2}+2x-1=0\Delta=b^{2}-4ac=2^{2}-4*3*(-1)=4+12=16\sqrt{Delta}=4\x_{1}=frac{-b-sqrt{Delta}}{2a}=frac{-2-4}{6}=frac{-6}{6}=-1\\x_{2}=frac{-b+sqrt{Delta}}{2a}=frac{-2+4}{6}=frac{2}{6}=frac{1}{3}\\oxed{xin {-1; frac{1}{3}}}[/latex] f) [latex](2x+1)^{2}=(4x+5)^{2}\4 x^{2} +4x+1=16 x^{2} +40x+25\16 x^{2} -4 x^{2} +40x-4x+25-1=0\12 x^{2} +36x+24=0 /:4\3 x^{2} +9x+6=0\Delta=b^{2}-4ac=9^{2}-4*3*6=81-72=9\sqrt{Delta}=3\x_{1}=frac{-b-sqrt{Delta}}{2a}=frac{-9-3}{6}=frac{-12}{6}=-2\\x_{2}=frac{-b+sqrt{Delta}}{2a}=frac{-9+3}{6}=frac{-6}{6}=-1\\oxed{xin {-2; -1}}[/latex]

Korzystamy ze wzoru skróconego mnożenia: [latex]a^{2}-b^{2}=(a+b)(a-b)[/latex] d) [latex]9x^{2}-(2x-1)^{2}=0 \ \ lbrack 3x-(2x-1)][3x+(2x-1)]=0 \ \ (3x-2x+1)(3x+2x-1)=0 \ \ (x+1)(5x-1)=0 \ \ x+1=0 vee 5x-1=0 \ \ x=-1 vee 5x=1 \ \ x=-1 vee x= frac{1}{5} \ \ oxed{xin lbrace -1, frac{1}{5} brace}[/latex] e) [latex]4x^{2}=(1-x)^{2} \ \ 4x^{2}-(1-x)^{2}=0 \ \ lbrack 2x-(1-x)][2x+(1-x)]=0 \ \ (2x-1+x)(2x+1-x)=0 \ \ (3x-1)(x+1)=0 \ \ 3x-1=0 vee x+1=0 \ \ 3x=1 vee x=-1 \ \ x= frac{1}{3} vee x=-1 \ \oxed{ xin lbrace -1, frac{1}{3}   brace}[/latex] f) [latex](2x+1)^{2}=(4x+5)^{2} \ \ (2x+1)^{2}-(4x+5)^{2} \ \ lbrack 2x+1-(4x+5)](2x+1+4x+5)=0 \ \ (2x+1-4x-5)(6x+6)=0 \ \ (-2x-4)(6x+6)=0 \ \ -2x-4=0 vee 6x+6=0 \ \ -2x=4 vee 6x=-6 \ \ x=-2 vee x=-1 \ \ oxed{xin lbrace -2,-1 brace}[/latex]