a) 4x^2-x^2+12x-3=0 3x^2+12x-3=0 Δ=144-4*3*(-3) Δ=144+36 Δ=180 √180=√10*√18=√10*3√2=3√20=3√4*√5=6√5 x1=(-12+6√5)/6=3(-4+2√5)/6=(-4+2√5)/2=-2+√5 x2=(-12-6√5)/6=3(-4-2√5)/6=-4-2√5/2=-2-√5 b) (6x^2+7x-3)(x^2-5x+6)=0 6x^2+7x-3=0 Δ=49-4*6*(-3) Δ=49+72 Δ=121 √121=11 x1=(-7-11)/12=-18/12=-1 i 1/2 x2=(-7+11)/12=4/12=2/6=1/3 x^2-5x+6=0 Δ=25-4*1*6 Δ=25-24 Δ=1 √1=1 x1=(5+1)/2=3 x2=(5-1)/2=2 c) (x+2)/(3x+5)=(3x+5)/(2x+4) 3x+5≠0 3x≠-5 x≠-5/3 2x+4≠0 2x≠-4 x≠-2 (2x+4)(x+2)=(3x+5)(3x+5) 2x^2+4x+4x+8=9x^2+15x+15x+25 2x^2+8x+8=9x^2+30x+25 /-25 2x^2+8x-17=9x^2+30x /-30x 2x^2-22x-17=9x^2 /-9x^2 -7x^2-22x-17=0 Δ=484-4*(-7)*(-17) Δ=484-476 Δ=8 √8=2√2 x1=(22-2√2)/-14=-2(-11+√2)/-14=(-11+√2)/7 x2=(22+2√2)/-14=-2(-11-√2)/-14=(-11-√2)/7
a) [latex]\4x^{2}-x^{2}+12x-3=0\ 3x^{2}+12x-3=0\ Delta=144+4*3*3=144+36=180\ sqrt{Delta}=6sqrt{5}\ x_{1}=frac{-12-6sqrt{5}}{6}=-2-sqrt{5}\ x_{2}=frac{-12+6sqrt{5}}{6}=-2+sqrt{5}[/latex] b) [latex]\(6x^{2}+7x-3)(x^{2}-5x+6)=0\\ Delta_{1}=49+4*6*3=121\ sqrt{Delta_{1}}=11\ x_{1}=frac{-7-11}{12}=frac{3}{2}\ x_{2}=frac{-7+11}{12}=frac{1}{3}\\ Delta_{2}=25-4*1*6=25-24=1\ sqrt{Delta_{2}}=1\ x_{3}=frac{-(-5)-1}{2}=frac{4}{2}=2\ x_{4}=frac{-(-5)+1}{2}=frac{6}{2}=3[/latex] c) [latex]frac{x+2}{3x+5}=frac{3x+5}{2x+4}\ 3x+5 eq0 wedge 2x+4 eq0\ x eq-frac{5}{3} wedge x eq-2\ D_{f}:R/[-2;-frac{5}{3}]\\ (x+2)(2x+4)=(3x+5)^{2}\ 2x^{2}+4x+4x+8=9x^{2}+30x+25\ 7x^{2}+22x+17=0\ Delta=484-4*7*17=484-476=8\ sqrt{Delta}=2sqrt{2}\ x_{1}=frac{-22-2sqrt{2}}{14}=frac{-11-sqrt{2}}{7}\ x_{2}=frac{-22+2sqrt{2}}{14}=frac{-11+sqrt{2}}{7}[/latex] Pozdrawiam :)
