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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{1}{40}x^2-\frac{1}{5}x+\frac{2}{5}=0\)
  2. \(\frac{4}{3}x^2+\frac{1}{3}x+\frac{25}{3}=0\)
  3. \(2x^2-(19x+60)=x(x-23)\)
  4. \(-(12+12x)=-x^2-(156-12x)\)
  5. \(2x^2-(11x-20)=x(x-2)\)
  6. \((3x-2)(3x+1)-x(7x-4)=-4\)
  7. \(\frac{9}{5}x^2+\frac{13}{10}x+\frac{1}{5}=0\)
  8. \(\frac{1}{5}x^2-3x+\frac{44}{5}=0\)
  9. \(x(x-31)=-45(x+1)\)
  10. \((-3x+4)(-4x-1)-x(11x-12)=-22\)
  11. \((-4x+3)(x-4)-x(-8x-40)=-133\)
  12. \((-x-3)(3x-5)-x(-5x+15)=33\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{1}{40}x^2-\frac{1}{5}x+\frac{2}{5}=0\\ \Leftrightarrow \color{red}{40.} \left(\frac{1}{40}x^2-\frac{1}{5}x+\frac{2}{5}\right)=0 \color{red}{.40} \\ \Leftrightarrow 4x^2-32x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-32x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-32)^2-4.4.64 & &\\ & = 1024-1024 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-32)}{2.4} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
  2. \(\frac{4}{3}x^2+\frac{1}{3}x+\frac{25}{3}=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{4}{3}x^2+\frac{1}{3}x+\frac{25}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow 16x^2+4x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+4x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.16.100 & &\\ & = 16-6400 & & \\ & = -6384 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  3. \(2x^2-(19x+60)=x(x-23) \\ \Leftrightarrow 2x^2-19x-60=x^2-23x \\ \Leftrightarrow x^2+4x-60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-60) & &\\ & = 16+240 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt256}{2.1} & & = \frac{-4+\sqrt256}{2.1} \\ & = \frac{-20}{2} & & = \frac{12}{2} \\ & = -10 & & = 6 \\ \\ V &= \Big\{ -10 ; 6 \Big\} & &\end{align} \\ -----------------\)
  4. \(-(12+12x)=-x^2-(156-12x) \\ \Leftrightarrow -12-12x=-x^2-156+12x \\ \Leftrightarrow x^2-24x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-24x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.1.144 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-24)}{2.1} & & \\ & = 12 & & \\V &= \Big\{ 12 \Big\} & &\end{align} \\ -----------------\)
  5. \(2x^2-(11x-20)=x(x-2) \\ \Leftrightarrow 2x^2-11x+20=x^2-2x \\ \Leftrightarrow x^2-9x+20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x+20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-9)^2-4.1.20 & &\\ & = 81-80 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-9)-\sqrt1}{2.1} & & = \frac{-(-9)+\sqrt1}{2.1} \\ & = \frac{8}{2} & & = \frac{10}{2} \\ & = 4 & & = 5 \\ \\ V &= \Big\{ 4 ; 5 \Big\} & &\end{align} \\ -----------------\)
  6. \((3x-2)(3x+1)-x(7x-4)=-4\\ \Leftrightarrow 9x^2+3x-6x-2 -7x^2+4x+4=0 \\ \Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
  7. \(\frac{9}{5}x^2+\frac{13}{10}x+\frac{1}{5}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{9}{5}x^2+\frac{13}{10}x+\frac{1}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 18x^2+13x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+13x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.18.2 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.18} & & = \frac{-13+\sqrt25}{2.18} \\ & = \frac{-18}{36} & & = \frac{-8}{36} \\ & = \frac{-1}{2} & & = \frac{-2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-2}{9} \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{1}{5}x^2-3x+\frac{44}{5}=0\\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2-3x+\frac{44}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2-15x+44=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-15x+44=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-15)^2-4.1.44 & &\\ & = 225-176 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-15)-\sqrt49}{2.1} & & = \frac{-(-15)+\sqrt49}{2.1} \\ & = \frac{8}{2} & & = \frac{22}{2} \\ & = 4 & & = 11 \\ \\ V &= \Big\{ 4 ; 11 \Big\} & &\end{align} \\ -----------------\)
  9. \(x(x-31)=-45(x+1) \\ \Leftrightarrow x^2-31x=-45x-45 \\ \Leftrightarrow x^2+14x+45=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+45=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.45 & &\\ & = 196-180 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-14-\sqrt16}{2.1} & & = \frac{-14+\sqrt16}{2.1} \\ & = \frac{-18}{2} & & = \frac{-10}{2} \\ & = -9 & & = -5 \\ \\ V &= \Big\{ -9 ; -5 \Big\} & &\end{align} \\ -----------------\)
  10. \((-3x+4)(-4x-1)-x(11x-12)=-22\\ \Leftrightarrow 12x^2+3x-16x-4 -11x^2+12x+22=0 \\ \Leftrightarrow x^2+11x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.18 & &\\ & = 121-72 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt49}{2.1} & & = \frac{-11+\sqrt49}{2.1} \\ & = \frac{-18}{2} & & = \frac{-4}{2} \\ & = -9 & & = -2 \\ \\ V &= \Big\{ -9 ; -2 \Big\} & &\end{align} \\ -----------------\)
  11. \((-4x+3)(x-4)-x(-8x-40)=-133\\ \Leftrightarrow -4x^2+16x+3x-12 +8x^2+40x+133=0 \\ \Leftrightarrow 4x^2+44x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+44x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (44)^2-4.4.121 & &\\ & = 1936-1936 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-44}{2.4} & & \\ & = -\frac{11}{2} & & \\V &= \Big\{ -\frac{11}{2} \Big\} & &\end{align} \\ -----------------\)
  12. \((-x-3)(3x-5)-x(-5x+15)=33\\ \Leftrightarrow -3x^2+5x-9x+15 +5x^2-15x-33=0 \\ \Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.(-18) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\ & = \frac{-18}{4} & & = \frac{8}{4} \\ & = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-08-03 19:02:50
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