Question

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Answer 1

2log₃(x-1) + log₃(x-5)² ≤ 2

                                                Z: x-1 > 0  => x > 1

                                                    x-5 ≠ 0  => x ≠ 5

                                                D: x > 1  i x ≠ 5

log₃(x-1)²+log₃(x-5)² ≤ log₃9

log₃[(x-1)²·(x-5)² ≤ log₃3² 

(x-1)²·(x-5)² ≤ 9

(x-1)²·(x-5)²-9 ≤ 0

(x²-2x+1)(x²-10x+25)-9 ≤ 0

x⁴-10x³+25x²-2x³+20x³-50x+x²-10x+25-9 ≤ 0

x⁴-12x³+46x²-60x+16 ≤ 0

 

x⁴-4x³-8x³+32x²+14x²-56x-4x+16 = 0

x³(x-4)-8x²(x-4)+14x(x-4)-4(x-4) = 0

(x-4)(x³-8x²+14x-4) = 0

(x-4)(x³-2x²-6x²+12x+2x-4) = 0

(x-4)[(x²(x-2)-6x(x-2)+2(x-2)] = 0

(x-4)(x-2)(x²-6x+2) = 0

     

x²-6x+2 = 0

Δ = b²-4ac = 36-8 = 28

√Δ = √(4·7) = 2√7 

x1 = (-b-√Δ)/2a = (6-2√7)/2 = 3-√7

x2 = (-b+√Δ)/2a = (6+2√7)/2 = 3+√7

x²-6x+2 = ((x-(3-√7))((x-(3+√7))

 

(x-4)*(x-2)*((x-(3-√7))*((x-(3+√7)) = 0

x = 4  v  x = 2  v  x = 3-√7  v  x = 3+√7 

x ∈ <3-√7; 2>  u  <4; 3+√7>   oraz D

 

Odp. x ∈ (1; 2>  u  <4; 5)  u  (5; 3+√7>

 

 

 

Answer 2

2log3 (x - 1) + log3 (x - 5)² ≤ 2        D: x > 1 i x ≠ 5 

log3 (x - 1)² + log3 (x - 5)² ≤ 2

log3 [ (x - 1)² * (x - 5)²] ≤ log3 3²

 (x - 1)² * (x - 5)² ≤  3²

(x² - 2x + 1)(x² - 10x + 25) - 9 ≤ 0

x⁴ - 10x³ + 25x² - 2x³ + 20x² - 50x + x² - 10x + 25 - 9 ≤ 0

x⁴ - 12x³ + 46x² - 60x + 16 ≤ 0

W(4) =4⁴ - 12 * 4³ + 46 * 4² - 60 * 4 + 16 = 256 - 768 + 736 - 240 + 16 = 0

 

x³ - 8x² + 14x - 4

-------------------------------

(x⁴ - 12x³ + 46x² - 60x + 16) : (x - 4)

-x⁴ + 4x³

------------------

     - 8x³ + 46x²

       8x³ - 32x²

      -----------------

                14x² - 60x

              - 14x² + 56x

                ---------------

                          - 4x + 16

                             4x - 16

                           ----------

                              =   =

Q(x) = x³ - 8x² + 14x - 4

Q(2) = 8 - 32 + 28 - 4 = 36 - 36 = 0

 

x² - 6x + 2

------------------------

(x³ - 8x² + 14x - 4) : (x - 2)

-x³ + 2x²

----------------

      - 6x² + 14x

        6x² - 12x

         ------------

                   2x - 4

                  -2x + 4

                  --------

                    =   =

x² - 6x + 2

Δ = 36 - 8 = 28

√Δ = √28 = 2√7

x1 = (6 - 2√7) / 2 = 3 - √7

x2 = (6 + 2√7) / 2 = 3 + √7

 

(x - 4) (x³ - 8x² + 14x - 4) ≤ 0

(x - 4)(x - 2)(x² - 6x + 2) ≤ 0

(x - 4)(x - 2)(x - x1)(x - x2) ≤ 0

   4         2        x1       x2

x ∈ <x1 ; 2> u < 4 ; x2 >     i   D

 

odp. x ∈ (1 ; 2> u < 4; 5) u (5, 3 + √7>