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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-(4-29x)=-x^2-(31-17x)\)
  2. \(\frac{3}{4}x=-\frac{1}{16}x^2-\frac{5}{4}\)
  3. \(x(16x-33)=-121(x+1)\)
  4. \(x(9x+15)=2(x-2)\)
  5. \((-3x-2)(-5x-2)-x(-3x+3)=12\)
  6. \(3x^2-(6x+18)=x(x-11)\)
  7. \(9x^2-(14x-2)=x(x-31)\)
  8. \(-(13-32x)=-2x^2-(5-17x)\)
  9. \((-5x+5)(4x+4)-x(-36x-10)=19\)
  10. \(x(4x+25)=5(x-5)\)
  11. \(2x^2-(20x-81)=x(x-2)\)
  12. \(-\frac{1}{10}x=-\frac{1}{5}x^2-\frac{4}{5}\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-(4-29x)=-x^2-(31-17x) \\ \Leftrightarrow -4+29x=-x^2-31+17x \\ \Leftrightarrow x^2+12x+27=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+27=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.27 & &\\ & = 144-108 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-12-\sqrt36}{2.1} & & = \frac{-12+\sqrt36}{2.1} \\ & = \frac{-18}{2} & & = \frac{-6}{2} \\ & = -9 & & = -3 \\ \\ V &= \Big\{ -9 ; -3 \Big\} & &\end{align} \\ -----------------\)
  2. \(\frac{3}{4}x=-\frac{1}{16}x^2-\frac{5}{4} \\ \Leftrightarrow \frac{1}{16}x^2+\frac{3}{4}x+\frac{5}{4}=0 \\ \Leftrightarrow \color{red}{16.} \left(\frac{1}{16}x^2+\frac{3}{4}x+\frac{5}{4}\right)=0 \color{red}{.16} \\ \Leftrightarrow x^2+12x+20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.20 & &\\ & = 144-80 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-12-\sqrt64}{2.1} & & = \frac{-12+\sqrt64}{2.1} \\ & = \frac{-20}{2} & & = \frac{-4}{2} \\ & = -10 & & = -2 \\ \\ V &= \Big\{ -10 ; -2 \Big\} & &\end{align} \\ -----------------\)
  3. \(x(16x-33)=-121(x+1) \\ \Leftrightarrow 16x^2-33x=-121x-121 \\ \Leftrightarrow 16x^2+88x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+88x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (88)^2-4.16.121 & &\\ & = 7744-7744 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-88}{2.16} & & \\ & = -\frac{11}{4} & & \\V &= \Big\{ -\frac{11}{4} \Big\} & &\end{align} \\ -----------------\)
  4. \(x(9x+15)=2(x-2) \\ \Leftrightarrow 9x^2+15x=2x-4 \\ \Leftrightarrow 9x^2+13x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+13x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.9.4 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.9} & & = \frac{-13+\sqrt25}{2.9} \\ & = \frac{-18}{18} & & = \frac{-8}{18} \\ & = -1 & & = \frac{-4}{9} \\ \\ V &= \Big\{ -1 ; \frac{-4}{9} \Big\} & &\end{align} \\ -----------------\)
  5. \((-3x-2)(-5x-2)-x(-3x+3)=12\\ \Leftrightarrow 15x^2+6x+10x+4 +3x^2-3x-12=0 \\ \Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  6. \(3x^2-(6x+18)=x(x-11) \\ \Leftrightarrow 3x^2-6x-18=x^2-11x \\ \Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.(-18) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\ & = \frac{-18}{4} & & = \frac{8}{4} \\ & = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
  7. \(9x^2-(14x-2)=x(x-31) \\ \Leftrightarrow 9x^2-14x+2=x^2-31x \\ \Leftrightarrow 8x^2+17x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+17x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.8.2 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.8} & & = \frac{-17+\sqrt225}{2.8} \\ & = \frac{-32}{16} & & = \frac{-2}{16} \\ & = -2 & & = \frac{-1}{8} \\ \\ V &= \Big\{ -2 ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
  8. \(-(13-32x)=-2x^2-(5-17x) \\ \Leftrightarrow -13+32x=-2x^2-5+17x \\ \Leftrightarrow 2x^2+15x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+15x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.2.(-8) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.2} & & = \frac{-15+\sqrt289}{2.2} \\ & = \frac{-32}{4} & & = \frac{2}{4} \\ & = -8 & & = \frac{1}{2} \\ \\ V &= \Big\{ -8 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  9. \((-5x+5)(4x+4)-x(-36x-10)=19\\ \Leftrightarrow -20x^2-20x+20x+20 +36x^2+10x-19=0 \\ \Leftrightarrow 16x^2+10x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+10x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.16.1 & &\\ & = 100-64 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt36}{2.16} & & = \frac{-10+\sqrt36}{2.16} \\ & = \frac{-16}{32} & & = \frac{-4}{32} \\ & = \frac{-1}{2} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
  10. \(x(4x+25)=5(x-5) \\ \Leftrightarrow 4x^2+25x=5x-25 \\ \Leftrightarrow 4x^2+20x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+20x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.4.25 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.4} & & \\ & = -\frac{5}{2} & & \\V &= \Big\{ -\frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
  11. \(2x^2-(20x-81)=x(x-2) \\ \Leftrightarrow 2x^2-20x+81=x^2-2x \\ \Leftrightarrow x^2-18x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-18x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-18)^2-4.1.81 & &\\ & = 324-324 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-18)}{2.1} & & \\ & = 9 & & \\V &= \Big\{ 9 \Big\} & &\end{align} \\ -----------------\)
  12. \(-\frac{1}{10}x=-\frac{1}{5}x^2-\frac{4}{5} \\ \Leftrightarrow \frac{1}{5}x^2-\frac{1}{10}x+\frac{4}{5}=0 \\ \Leftrightarrow \color{red}{10.} \left(\frac{1}{5}x^2-\frac{1}{10}x+\frac{4}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 4x^2-2x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-2x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.4.16 & &\\ & = 4-256 & & \\ & = -252 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
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