[tex]zad.1\\\\x^{2} -9\neq 0~~\Rightarrow ~~x\neq 3,~~x\neq -3 \\D=R- \{-3,3\}\\\\\dfrac{x^{2}-3x }{x^{2} -9} =\dfrac{x\cdot (x-3)}{(x-3)\cdot (x+3)} =\dfrac{x}{x+3}[/tex]
[tex]zad.2\\\\4x^{2} -1\neq 0~~\Rightarrow ~~x\neq \dfrac{1}{2} ,~~x\neq -\dfrac{1}{2}\\\\D=R- \{-\dfrac{1}{2},\dfrac{1}{2} \}\\\\\\\dfrac{4x^{2} +8x+3}{4x^{2} -1} =\dfrac{4(x+\frac{1}{2})(x+\frac{3}{2}) }{4(x^{2}-\frac{1}{4} ) } =\dfrac{(x+\frac{1}{2})(x+\frac{3}{2}) }{(x^{2}-\frac{1}{4} ) } = \dfrac{(x+\frac{1}{2})(x+\frac{3}{2}) }{(x-\frac{1}{2} )\cdot (x+\frac{1}{2}) } =\dfrac{(x+\frac{3}{2})}{(x-\frac{1}{2})}[/tex]
[tex]4x^{2} +8x+3=4(x+\frac{1}{2})(x+\frac{3}{2} )\\\Delta=84-64=16\\\sqrt{\Delta}=4\\x_{1}=\frac{-8-4}{8} =-\frac{3}{2} \\\lor\\ x_{2}=\frac{-8+4}{8} =-\frac{1}{2} \\\\[/tex]
[tex]zad.3\\\\x^{2} +x-6\neq 0~~\Rightarrow~~(x+3)(x-2)\neq 0~~\Rightarrow ~~x\neq-3,~~x\neq 2~~\Rightarrow ~~D=R- \{-3,2 \} \\\Delta=1-4\cdot 1\cdot (-6)=25\\\sqrt{\Delta}=5\\x_{1} =\frac{-1-5}{2} =-3\\\lor\\x_{2} =\frac{-1+5}{2} =2\\\\\dfrac{x^{2} -x-2}{x^{2} +x-6} =\dfrac{(x-2)(x+1)}{(x+3)(x-2)} =\dfrac{x+1}{x+3} \\\\x^{2} -x-2=(x-3)(x+1)\\\Delta=1+8=9\\\sqrt{\Delta} =3\\x_{1}=2~~\lor ~~x_{2} =-1[/tex]
[tex]zad.4\\\\x^{2} -6x+9\neq 0~~(x-3)^{2} \neq 0~~\Rightarrow ~~x\neq 3 ~~\Rightarrow ~~D=R - \{3 \}\\\\\dfrac{x^{2}-25 }{x^{2} -6x+9} \div \dfrac{x+5}{x-3} =\dfrac{(x-5)(x+5)}{(x-3)^{2}} \cdot \dfrac{x-3}{x+5} =\dfrac{(x-5)(x+5)}{(x-3)(x-3)} \cdot \dfrac{x-3}{x+5} =\dfrac{x-5}{x-3}[/tex]
[tex]zad.5\\\\4x^{2} -9\neq 0,~~(2x-3)(2x+3)\neq 0~~\Rightarrow ~~x\neq \dfrac{3}{2} ,~~x\neq -\dfrac{3}{2}\\\\3x^{2} -9x\neq 0,~~3x(x-3)\neq 0~~\Rightarrow ~~x\neq 0,~~x\neq 3\\\\D=R- \{ -\dfrac{3}{2} ,~~0,~~\dfrac{3}{2} ,~~3 \}\\\\\dfrac{6x^{2} }{4x^{2}-9 } \cdot \dfrac{2x+3}{3x^{2}-9x } =[/tex]
[tex]=\dfrac{6\cdot x \cdot x}{(2x-3)(2x+3)} \cdot \dfrac{2x+3}{3x(x-3)} =\dfrac{2x}{(2x-3)(x-3)} =\dfrac{2x}{2x^{2}-6x-3x+9 } =\dfrac{2x}{2x^{2}-9x+9 }[/tex]
