Rozwiąż równanie. a)x²-4x-5=0 Δ = b² - 4ac = (-4)² - 4 *1*(-5) = 16 + 20 = 36 √Δ = √36 = 6 x₁ = (-b-√Δ)/2a = ( 4 -6)/2 = -2/2 = -1 x₂ = (-b+√Δ)/2a = ( 4 +6)/2 = 10/2 = 5 b)3x²-14x-5=0 Δ = b² - 4ac = (-14)² - 4 *3*(-5) = 196 + 60 = 256 √Δ = √256 =16 x₁ = (-b-√Δ)/2a = ( 14 -16)/6 = -2/6 = -1/3 x₂ = (-b+√Δ)/2a = ( 14 +16)/2 = 30/2 = 15 c)4x²-2x+¼=0 /*4 16x² - 8x + 1 = 0 Δ = b² - 4ac = (-8)² - 4 *16*1 = 64 -64 = 0 x₁ = x₂ = -b/2a = 8/2 = 4 d)5x²+2x+1=0 Δ = b² - 4ac = 2² - 4 *5*1 = 4 – 20 = -16 Brak rozwiązania bo delta ujemna e)7x²+2x=1 7x² + 2x – 1 = 0 Δ = b² - 4ac =2² - 4 *7*(-1) = 4 + 28 = 32 √Δ = √32 =√1 6*2 = 4√2 x₁ = (-b-√Δ)/2a = ( -2 -4√2)/14 = -2(1+2√2)/14 = - (1+ 2√2)/7 x₂ = (-b+√Δ)/2a = ( -2 +4√2)/14 = 2(2√2 -1)/14 = ( 2√2 + 1)/7 f)-6x²+x+1=0 Δ = b² - 4ac = 1² - 4 *(-6)*1 = 1 + 24 = 25 √Δ = √25 = 5 x₁ = (-b-√Δ)/2a = ( -1 -5)/-12 = -6/12 = 1/2 x₂ = (-b+√Δ)/2a = ( -1 +5)/-12 = - 4/12 = - 1/3 2.Rozwiąż rónanie. a)x(x-5)=2x(x-1) x² - 5x = 2x² - 2x x² - 5x -2x² + 2x = 0 -x² - 3x = 0 -x (x – 3) = 0 x = 0 lub x-3 = 0 x = 0 lub x = 3 b)2(1-5x)=(1-x)² 2 – 10x = 1 – 2x - x² 2 – 10x -1 + 2x + x² = 0 x² - 8x + 1 = 0 Δ = b² - 4ac = (-8)² - 4 *1* = 64 -4 =60 √Δ = √60 = √4*15 = 2√15 x₁ = (-b-√Δ)/2a = ( 8 -2√15)/2 = 2(4 -√15)/2 = 4 -√15 x₂ = (-b+√Δ)/2a = ( 8 +2√15)/2 = 2(4 +√15)/2 = 4 +√15 c)x²+4/5=1-x/2 2(x²+4) = 5(1-x) 2x² + 8 = 5 – 5x 2x² + 5x + 3 = 0 Δ = b² - 4ac = 5² - 4 *2*3= 25- 24 = 1 √Δ = √1 = 1 x₁ = (-b-√Δ)/2a = ( -5 -1)/4 = -6/4 = - 1 1/2 x₂ = (-b+√Δ)/2a = ( -5 +1)/4 = -4/4 = - 1 d)(x+1)²/3=(x+4)²/7- 1 /* 21 7(x+1)²=3(x+4)² - 21 7( x² + 2x + 1) = 3(x² + 8x + 16) -21 7x² + 14x + 7 = 3x² + 24x + 48 -21 7x² + 14x + 7 - 3x² - 24x- 48 + 21 = 0 4x² - 10x –20 = 0 Δ = b² - 4ac = (-10)² - 4 *4*(-20)= 100 + 320= 420 √Δ = √420 =√4*105 = 2√105 x₁ = (-b-√Δ)/2a = (10 -2√105)/8 = 2(5 -√105)/8 = (5 -√105)/4 x₂ = (-b+√Δ)/2a = (10 +2√105)/8 = 2(5 +√105)/8 = (5 +√105)/4
