a) [latex]3x^2 + 2x-2 = 0[/latex] [latex]Delta = 2^2 - 4 cdot 3 cdot (-2) = 4 + 24 = 28[/latex] [latex]sqrt{Delta} = sqrt{28} = sqrt{4 cdot 7} = 2sqrt{7}[/latex] [latex]x_1 = frac{-2-2sqrt{7}}{2 cdot 3}= frac{-1-sqrt{7}}{3}[/latex] [latex]x_2 = frac{-2+2sqrt{7}}{2 cdot 3}= frac{-1+sqrt{7}}{3}[/latex] b) [latex]2-4x=3x^2[/latex] [latex]-3x^2-4x+2=0[/latex] [latex]Delta = (-4)^2 - 4 cdot (-3) cdot 2= 16 + 24 = 40[/latex] [latex]sqrt{Delta} = sqrt{40} = sqrt{4 cdot 10} = 2sqrt{10}[/latex] [latex]x_1 = frac{4-2sqrt{10}}{2 cdot (-3)}= frac{-2+sqrt{10}}{3}[/latex] [latex]x_2 = frac{4+2sqrt{10}}{2 cdot (-3)}= frac{-2-sqrt{10}}{3}[/latex] [latex]frac{x}{2} = frac{3 cdot (5x-1)^2}{5}[/latex] [latex]5x =6 cdot (5x-1)^2[/latex] [latex]5x =6 cdot (25x^2 - 10x + 1)[/latex] [latex]150x^2 - 60x + 6 = 5x[/latex] [latex]150x^2 - 60x + 6 - 5x=0[/latex] [latex]150x^2 - 65x + 6 = 0[/latex] [latex]Delta = (-65)^2 - 4 cdot 150 cdot 6 = 4225 - 3600= 625[/latex] [latex]sqrt{Delta} = sqrt{625} =25[/latex] [latex]x_1 = frac{65 - 25}{2 cdot 150} = frac{40}{300} = frac{2}{15}[/latex] [latex]x_2 = frac{65 +25}{2 cdot 150} = frac{90}{300} = frac{3}{10}[/latex] Zatem rówanie [latex]frac{x}{2} = frac{3 cdot (5x-1)^2}{5}[/latex] możemy zapisać w postaci: [latex]150 cdot (x - frac{2}{15})(x - frac{3}{10}) = 0[/latex] [latex]15 cdot (x - frac{2}{15}) cdot 10 cdot (x - frac{3}{10}) = 0[/latex] [latex](15x - 2)(10x - 3) = 0[/latex] x, y - szukane liczby {x + y = 2 {x·y = - 1 {x = 2 - y {(2 - y)·y = - 1 {x = 2 - y {2y - y² = - 1 {x = 2 - y {- y² + 2y + 1 = 0 Rozwiążemy drugie równanie - y² + 2y + 1 = 0 Δ = 2² - 4 · (- 1) · 1 = 4 + 4 = 8 √Δ = √8 = √4·2 = 2√2 y₁ = (- 2 - 2√2) / [2 · (- 1)] = (- 2 - 2√2) / (- 2) = 1 + √2 y₂ = (- 2 + 2√2) / [2 · (- 1)] = (- 2 + 2√2) / (- 2) = 1 - √2 {x₁ = 2 - y₁ {y₁ = 1 + √2 {x₁ = 2 - (1 + √2) {y₁ = 1 + √2 {x₁ = 2 - 1 - √2 {y₁ = 1 + √2 {x₁ = 1 - √2 {y₁ = 1 + √2 {x₂ = 2 - y₂ {y₂ = 1 - √2 {x₂ = 2 - (1 - √2) {y₂ = 1 - √2 {x₂ = 2 - 1 + √2 {y₂ = 1 - √2 {x₂ = 1 + √2 {y₂ = 1 - √2 Odp. Szukane liczby to 1 - √2 i 1 + √2.
1a] 3x²+2x-2=0 Δ=b²-4ac=4+24=28 √Δ=2√7 x₁=[-b-√Δ]/2a=[-2-2√7]/6=(-1-√7)/3 x₂=[-b+√Δ]/2a=[-2+2√7]/6=(√7-1]/3 b] 2-4x=3x² 3x²+4x-2=0 Δ=16+24=40 √Δ=2√10 x₁=[-4-2√10]/6=(-2-√10)/2 x₂=[-4+2√10]/6=(√10-2]/3 2] x/2=3(5x-1)²/5 5x=6(25x²-10x+1) 5x=150x²-60x+6 150x²-65x+6=0 Δ=4225-3600=625 √Δ=25 x₁=[65-25]/300=40/300=²/₁₅ x₂=[65+25]/300=90/300=³/₁₀ postać iloczynowa; a(x-x₁)(x-x₂)=0 150(x-²/₁₅)(x-³/₁₀)=0 15(x-²/₁₅)×10(x-³/₁₀)=0 (15x-2)(10x-3)=0 3] a,b=szukane liczby a+b=2 ab=-1 b=2-a a(2-a)=-1 2a-a²=-1 a²-2a-1=0 Δ=4+4=8 √Δ=2√2 a₁=[2-2√2]/2=1-√2 a₂=[2+2√2]/2=1+√2 b₁=2-1+√2=1+√2 b₂=2-1-√2=1-√2 szukane liczby to ; 1-√2 i 1+√2
