Matematyka
wera1014
2017-06-21 13:38:36
Rozwiąż równania wykładnicze: a. [latex] (frac{4}{3} )^{ x^{2} +5x} = ( frac{9}{16} ) ^{x-2} * (0,75)^{ x^{2} } [/latex] b. [latex] (frac{2}{5} )^{ x^{2}-1} * (frac{25}{4})^{-2x} = ( sqrt{ frac{2}{5} } )^{4x-4 x^{2} } [/latex]
Odpowiedź
laleczka199714
2017-06-21 14:12:24

[latex]a. \ (frac{4}{3} )^{ x^{2} +5x} = ( frac{9}{16} ) ^{x-2} * (0,75)^{ x^{2} } \ (frac{4}{3} )^{ x^{2} +5x} = [( frac{4}{3} )^{-2}] ^{x-2} * (frac{3}{4})^{ x^{2} } \ (frac{4}{3} )^{ x^{2} +5x} =( frac{4}{3} )^{-2x+4} * (frac{4}{3})^{- x^{2} } \ x^2-5x=-2x+4-x^2\ 2x^2-3x-4=0\ Delta = b^2-4ac=(-3)^2-4cdot 2cdot (-4)=9+16=25\ x_{1}=frac{-b-sqrt{Delta}}{2a}=frac{3-5}{2cdot 2}=frac{-2}{4}=-frac{1}{2}\x_{2}=frac{-b+sqrt{Delta}}{2a}=frac{3+5}{2cdot 2}=frac{8}{4}=2\ \ [/latex] [latex]xin {-frac{1}{2},2}[/latex] [latex]b. \ (frac{2}{5} )^{ x^{2}-1} * (frac{25}{4})^{-2x} = ( sqrt{ frac{2}{5} } )^{4x-4 x^{2} } \ (frac{2}{5} )^{ x^{2}-1} * [ (frac{2}{5})^{-2}]^{-2x} = [( frac{2}{5} )^{frac{1}{2}}]^{4x-4 x^{2} } \ (frac{2}{5} )^{ x^{2}-1} * [ (frac{2}{5})^{4x} = [( frac{2}{5} )^{2x-2 x^{2} } \ x^2-1+4x=2x-2x^2\ 3x^2+2x-1=0\ Delta=2^2-4cdot 3cdot (-1)=4+12=16\ x_{1}=frac{-2-4}{2cdot 3}=frac{-6}{6}=-1\ x_{2}=frac{-2+4}{2cdot 3}=frac{2}{6}=frac{1}{3}\ \ xin {-1, frac{1}{3}} [/latex]

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