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3 do potęgi 2x - 2 * 3do potęgi (x-1) - 7 = 0
3 do potęgi x =t, t>0
t²-2/3 t-7=0 /*3
3t² -2t-21=0
Δ=4+252=256, √Δ=16
t₁=-14/6 odpada
t₂=3
3 do potęgi x =3
x=1
Zadanie 2. Rozwiąż nierówności
a)
5do potęgi (x ² - 7x + 12) > 1
5do potęgi (x ² - 7x + 12) >5⁰
x ² - 7x + 12>0
Δ=49-48=1, √Δ=1
x₁=3
x₂=4
x∈(-∞,3)u(4,+∞)
b)
log₃(x² + 2) > 3
log₃(x² + 2) > log₃3³
x² + 2>3³
x² + 2>27
x² -25>0
(x-5)(x+5)>0
x=5, x=-5
x∈(-∞,-5)u(5,+∞)
(3^x)²-⅔(3^x)-7=0 / *3
3(3^x)²-2(3^x)-21=0
(3^x)=t;t>0
3t²-2t-21=0
Δ=(-2)²-4*3*(-21)=256
√Δ=16
t=2-16/6=-(14/6)<0, odpada
t=2+16/6=3
3^x=3
x=1
a)
5^(x²-7x+12)>5⁰
x²-7x+12>0
Δ=(-7)²-4*1*12=1
√Δ=1
x₁=3 x₂=4
x∈(-∞,3) i (4,+∞)
b) log₃(x² + 2) > 3
log₃(x²+2)>log₃27
x²+2>27
x²>25
x∈(-∞,-5)i(5,+∞)