Question

Sa se rezolve ecuatiile exponențiale
Din poza de mai jos

Answer 1

a) 10^(x^2-3x) = 1/100 <-> 10^(x^2-3x)=10^(-2)  -> x^2-3x= -2 -> x^2-3x+2=0
delta=b^-4*a*c
delta=(-3)^2-4*1*2=1
x1=(-b+radical din  delta)/2*1=3+1/2=2
x2=(-b-radical din delta)/2*1=3-1/2=1
b) 5^(x^2+1)=3125 <-> 5^(x^2+1)=5^5  -> x^2+1=5 -> x^2=4 -> x=2 sau x= -2
c) 2^(x^2-8x+4,5)=16* radical din 2 <-> 2^(x^2-8x+4,5)=2^4*2^1/2 <-> 2^(x^2-8x+4,5)=2^9/2 <-> x^2-8x+4,5=4,5 <-> x(x-8)=0 , deci x=0 sau x-8=0 -> x=8
d) 4^(radical din x+1)=8 <-> 2^2*radical din x+1=2^3 <-> 2*radical din x+1=3 <-> radical din x+1=3/2 <-> x+1=9/4 -> x= 9/4-4/4 -> x=5/4

Answer 2

a)
10ˣ^² ⁻³ˣ =10⁻²
x²-3x= -2
x²-3x+2=0
x-2x-x+2=0
x(x-2)- (x-2)=0
(x-2)(x-1)=0
x₁=2;    x₂=1

b)
5ˣ^² ⁺¹ = 3125
5ˣ^² ⁺¹ = 5⁵
x²+1-5=0
x²-4=0
(x+2)(x-2)=0
x₁= -2; x₂= 2

c)
2ˣ^²- ⁸ˣ⁺⁴₎⁵= 16*√2
2ˣ^²- ⁸ˣ⁺⁴₎⁵= 2⁴* 2^1/2
x²- 8x +4,5= 4+1/2
x²- 8x +4,5= 8/2+1/2
x²- 8x +9/2= 9/2
x²- 8x = 0
x(x-8)=0
x₁=0
x₂=8

d) 4^(√(x+1)=8
2²^√(x+1)= 2³
2√(x+1)=3
√(x+1) =3/2
√(x+1) =1,5
√(x+1) ² =1,5²
x+1= 2,25
x=2,25-1
x=1,25