4. a) Δ=(4)² - 4 * 1 * (-21) = 16 + 84 = 100 √Δ= √100 = 10 x₁= ⁻⁴ ⁻ ¹°°/₂*₁ = ⁻¹°⁴/₂= - 52 x₂= ⁻⁴ ⁺ ¹°°/₂ = ⁹⁶/₂ = 48 b) -x² - x + 6= 0 Δ = 1 - 4 * (-1) * 6 = 1 + 24 = 25 √Δ= 5 x₁= ¹ ⁻ ⁵/₋₂ = ⁴/₋₂= -2 x₂= ¹ ⁺ ⁵/₋₂= ⁶/₋₂= -3 c) -x² +x + 12=0 Δ=1-4*(-1)*12=1+48=49 √Δ=7 x₁= ⁻¹⁻⁷/₋₂=4 x₂= ⁻¹⁺⁷/₋₂= -3 d) Δ= 25-4*2*(-3)= 25+24=49 √Δ=7 x₁= ⁻⁵⁻⁷/₂*₂₅= ⁻¹²/₅₀=⁻⁶/₂₅ x₂=⁻⁵⁺⁷/₅₀=¹/₂₅ e) Δ= 49-4*(-6)*5=49+120=169 √Δ=13 x₁ = ⁷⁻¹³/₂*₄₉=⁻⁶/₉₈=⁻³/₄₉ x₂=⁷⁺¹³/₉₈=¹⁰/₄₉ f) Δ=144-4*⁹/₄*16=144-144=0 x= ¹²/₂*₉/₄= ⁸/₃
W kolejności: * pierwsze zdjęcie a) [latex]x^{2}+4x-21=0 \ Delta=16+4*1*21=100 \ sqrt{Delta}=10 \ x_{1}= frac{-4-10}{2} = frac{-14}{2} =-7 \ x_{2}= frac{-4+10}{2} = frac{6}{2} =3[/latex] b) [latex]6-x-x^{2}=0 |*(-1) \ x^{2}+x-6=0 \ Delta=1-4*1*(-6)=1+4*6=1+24=25 \ sqrt{Delta}=5 \ x_{1}= frac{-1-5}{2} = frac{-6}{2} =-3 \ x_{2}= frac{-1+5}{2} = frac{4}{2} =2[/latex] c) [latex]x-x^{2}+12=0|*(-1) \ x^{2}-x-12=0 \ Delta=1-4*1*(-12)=1+4*12=1+48=49 \ sqrt{Delta}=7 \ x_{1}= frac{1-7}{2} = frac{-6}{2} =-3 \ x_{2}= frac{1+7}{2} = frac{8}{2} =4[/latex] d) [latex]2x^{2}+5x-3=0 \ Delta=25-4*2*(-3)=25+4*2*3=25+24=49 \ sqrt{Delta}=7 \ x_{1}= frac{-5-7}{4} = frac{-12}{4} =-3 \ x_{2}= frac{-5+7}{4} = frac{2}{4} = frac{1}{2} [/latex] e) [latex]-6x^{2}-7x+5=0|*(-1) \ 6x^{2}+7x-5=0 \ Delta=49-4*6*(-5)=49+4*6*5=49+120=169 \ sqrt{Delta}=13 \ x_{1}= frac{-7-13}{12} = frac{-20}{12} = frac{-5}{3} \ x_{2}= frac{-7+13}{12} = frac{6}{12} = frac{1}{2} [/latex] f) [latex] frac{9}{4} x^{2}-12x+16=0 \ Delta=144-4* frac{9}{4}*16=144-144=0 \ x_{0}= frac{12}{2* frac{9}{4} } = frac{12}{ frac{9}{2} } =12* frac{2}{9} = frac{8}{3} [/latex] ** drugie zdjęcie (w miejsce przecinków wstaw "lub") Korzystam tutaj ze wzoru skróconego mnożenia: [latex]a^{2}-b^{2}=(a-b)(a+b)[/latex] a) [latex]x^{2}-49=0 \ (x-7)(x+7)=0 \ x-7=0,...x+7=0\ x=7.....x=-7[/latex] b) [latex]x^{2}-144=0 \ (x-12)(x+12)=0 \ x-12=0,...x+12=0 \ x=12.....x=-12[/latex] c) [latex]9x^{2}-1=0 \ (3x-1)(3x+1)=0 \ 3x-1=0,...3x+1=0 \ 3x=1...3x=-1 \ x= frac{1}{3}......x= -frac{1}{3}[/latex] d) [latex]36x^{2}-1=0 \ (6x-1)(6x+1)=0 \ 6x-1=0,...6x+1=0 \ 6x=1.....6x=-1 \ x=frac{1}{6}.......x=- frac{1}{6}[/latex] e) [latex]4x^{2}-81=0 \ (2x-9)(2x+9)=0 \ 2x-9=0,...2x+9=0 \ 2x=9....2x=-9 \ x=frac{9}{2}.....x=- frac{9}{2}[/latex] f) [latex]25x^{2}-36=0 \ (5x-6)(5x+6)=0 \ 5x-6=0, 5x+6=0 \ 5x=6....5x=-6 \ x=frac{6}{5}.....x=- frac{6}{5}[/latex] *** trzecie zdjęcie, tutaj trochę bardziej problematyczne przykłady, ale damy radę ;-) a) [latex]-7x^{2}+42x=0 \ 42x-7x^{2}=0 \ x(42-7x)=0 \ x=0,...42-7x=0 \ ..............42=7x \ ...............x= frac{42}{7} \ ...............x=6[/latex] b) [latex]9x^{2}+ frac{1}{3} x=0 \ x(9x+frac{1}{3})=0 \ x=0,...9x+frac{1}{3}=0 \ .............9x=-frac{1}{3} \ ...............x=-frac{1}{27}[/latex] c) [latex]36x^{2}=6x \ 36x^{2}-6x=0 \ 6x(6x-1)=0 \ 6x=0,...6x-1=0 \ x=0.......6x=1 \ ................x=frac{1}{6}[/latex] d) [latex]2x+18x^{2}=0 \ 2x(1+9x)=0 \ 2x=0,...1+9x=0 \ x=0......9x=-1 \ ................x=-frac{1}{9}[/latex]
