3 zadania w załączniku dotyczące wyrażeń wymiernych.

Zadanie 4 [latex]a)frac{2x-10}{-3x^2+9x+30}=* \---------- \-3x^2+9x+30=0 \x^2-3x-10=0 \Delta=9+40=49 Rightarrow sqrt{Delta}=7 \x_1=frac{3-7}{2}=-2 \x_2=frac{3+7}{2}=5 \--------- \ *=frac{2(x-5)}{-3(x+2)(x-5)}=-frac{2}{3(x+2)} \x eq -2[/latex] [latex]b)frac{2x^2+sqrt{3}x-3}{3x^2-9}=* \---------- \2x^2+sqrt{3}x-3=0 \Delta=3+24=37 Rightarrow sqrt{Delta}=3sqrt{3} \x_1=frac{-sqrt{3}-3sqrt{3}}{4}=-sqrt{3} \x_2=frac{sqrt{3}}{2} \---------- \ *=frac{2(x+sqrt{3})(x-frac{sqrt{3}}{2})}{3(x-sqrt{3})(x+sqrt{3})}=frac{2x-sqrt{3}}{3x-3sqrt{3}} \x eq sqrt{3} [/latex] [latex]c)frac{x^2-5x+6}{x^2-4x+4}=* \---------- \x^2-5x+6=0 \Delta=25-24=1=sqrt{Delta} \x_1=frac{5-1}{2}=2 \x-2=frac{5+1}{2}=3 \---------- \=frac{(x-2)(x-3)}{(x-2)^2}=frac{x-3}{x-2} \x eq 2[/latex] [latex]d)frac{(x-2)^2-9}{(x+3)^2-4}=frac{(x-2-3)(x-2+3)}{(x+3-2)(x+3+2)}=frac{(x-5)(x+1)}{(x+1)(x+5)}=frac{x-5}{x+5} \x eq -5[/latex] ________________________________ Zad6 [latex]a)frac{6x^2+x}{x^3+x^2}=frac{x(6x+1)}{x(x^2+x)}=frac{6x+1}{x^2+x} \Dla quad x=-sqrt{3} \frac{6x+1}{x^2+x}=frac{-6sqrt{3}+1}{3-sqrt{3}}=frac{-18sqrt{3}+3-18+sqrt{3}}{9-3}=frac{-15-17sqrt{3}}{6}[/latex] [latex]b)frac{3x-sqrt{2}}{9x^2-2}=frac{3x-sqrt{2}}{(3x-sqrt{2})(3x+sqrt{2})}=frac{1}{3x+sqrt{2}} \Dla quad x=-sqrt{2} \frac{1}{3x+sqrt{2}}=frac{1}{-3sqrt{2}+sqrt{2}}=-frac{1}{2sqrt{2}}=-frac{sqrt{2}}{4}[/latex] [latex]c)frac{sqrt{5}x}{5x^2-sqrt{5}x}=frac{sqrt{5}x}{x(5x-sqrt{5})}=frac{sqrt{5}}{5x-sqrt{5}} \Dla quad x=2sqrt{5} \frac{sqrt{5}}{5x-sqrt{5}}=frac{sqrt{5}}{10sqrt{5}-sqrt{5}}=frac{sqrt{5}}{9sqrt{5}}=frac{1}{9}[/latex] [latex]d)frac{5x^5-x^3}{sqrt{125}x^6-sqrt{5}x^4}=frac{x^3(5x^2-1)}{sqrt{5}x^4(5x^2-1)}=frac{1}{sqrt{5}x} \Dla quad x=-frac{sqrt{5}}{5} \frac{1}{sqrt{5}x}=frac{1}{sqrt{5}}cdot (-frac{5}{sqrt{5}})=-frac{5}{5}=-1[/latex] ________________________________ Zadanie 7 [latex]a)frac{(50x^2-18)^2}{(3+5x)(5x-3)}=frac{left(2(5 x-3)(5 x+3) ight )^2}{(5x+3)(5x-3)}=4(25x^2-9)=100x^2-36 \Dla quad x=-frac{1}{5} \100x^2-36=100cdot frac{1}{25}-36=4-36=-32[/latex] [latex]b)frac{(-2x-3y)^2-25x^2}{7(x+frac{3}{7}y)^2}=frac{(-2x-3y-5x)(-2x-3y+5x)}{7(x+frac{3}{7}y)^2}=frac{(-7x-3y)(3x-3y)}{7(x+frac{3}{7}y)^2}= \=frac{-7(x+frac{3}{7}y)(3x-3y)}{7(x+frac{3}{7}y)^2}=frac{3(y-x)}{(x+frac{3}{7}y)} \Dla quad x=9sqrt{7}quad y=4sqrt{7} \frac{3(y-x)}{(x+frac{3}{7}y)}=frac{3(4sqrt{7}-9sqrt{7})}{9sqrt{7}+frac{12sqrt{7}}{7}}=frac{-15sqrt{7}}{frac{75}{7}sqrt{7}}=-frac{7}{5}[/latex]