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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{1}{48}x^2+\frac{1}{3}x+\frac{4}{3}=0\)
  2. \((-4x-1)(x+2)-x(-8x+10)=-27\)
  3. \(x(9x-5)=-(x+1)\)
  4. \(-(9-14x)=-12x^2-(6-9x)\)
  5. \(\frac{17}{2}x=-\frac{1}{2}x^2-35\)
  6. \((3x+3)(4x-1)-x(11x+10)=-58\)
  7. \((4x-5)(x+5)-x(-8x-30)=-37\)
  8. \(\frac{2}{3}x=-\frac{1}{15}x^2-\frac{5}{3}\)
  9. \(3x^2+\frac{7}{6}x-\frac{4}{3}=0\)
  10. \(\frac{1}{5}x^2+\frac{5}{2}x+\frac{36}{5}=0\)
  11. \(2x^2-(15x+32)=x(x-19)\)
  12. \(-(14+43x)=-9x^2-(95-11x)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{1}{48}x^2+\frac{1}{3}x+\frac{4}{3}=0\\ \Leftrightarrow \color{red}{48.} \left(\frac{1}{48}x^2+\frac{1}{3}x+\frac{4}{3}\right)=0 \color{red}{.48} \\ \Leftrightarrow x^2+16x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.64 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.1} & & \\ & = -8 & & \\V &= \Big\{ -8 \Big\} & &\end{align} \\ -----------------\)
  2. \((-4x-1)(x+2)-x(-8x+10)=-27\\ \Leftrightarrow -4x^2-8x-x-2 +8x^2-10x+27=0 \\ \Leftrightarrow 4x^2-20x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-20x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-20)^2-4.4.25 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-20)}{2.4} & & \\ & = \frac{5}{2} & & \\V &= \Big\{ \frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
  3. \(x(9x-5)=-(x+1) \\ \Leftrightarrow 9x^2-5x=-x-1 \\ \Leftrightarrow 9x^2-4x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-4x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.9.1 & &\\ & = 16-36 & & \\ & = -20 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  4. \(-(9-14x)=-12x^2-(6-9x) \\ \Leftrightarrow -9+14x=-12x^2-6+9x \\ \Leftrightarrow 12x^2+5x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+5x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.12.(-3) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.12} & & = \frac{-5+\sqrt169}{2.12} \\ & = \frac{-18}{24} & & = \frac{8}{24} \\ & = \frac{-3}{4} & & = \frac{1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{17}{2}x=-\frac{1}{2}x^2-35 \\ \Leftrightarrow \frac{1}{2}x^2+\frac{17}{2}x+35=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+\frac{17}{2}x+35\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2+17x+70=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+17x+70=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.1.70 & &\\ & = 289-280 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt9}{2.1} & & = \frac{-17+\sqrt9}{2.1} \\ & = \frac{-20}{2} & & = \frac{-14}{2} \\ & = -10 & & = -7 \\ \\ V &= \Big\{ -10 ; -7 \Big\} & &\end{align} \\ -----------------\)
  6. \((3x+3)(4x-1)-x(11x+10)=-58\\ \Leftrightarrow 12x^2-3x+12x-3 -11x^2-10x+58=0 \\ \Leftrightarrow x^2-16x+55=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+55=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.55 & &\\ & = 256-220 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-16)-\sqrt36}{2.1} & & = \frac{-(-16)+\sqrt36}{2.1} \\ & = \frac{10}{2} & & = \frac{22}{2} \\ & = 5 & & = 11 \\ \\ V &= \Big\{ 5 ; 11 \Big\} & &\end{align} \\ -----------------\)
  7. \((4x-5)(x+5)-x(-8x-30)=-37\\ \Leftrightarrow 4x^2+20x-5x-25 +8x^2+30x+37=0 \\ \Leftrightarrow 12x^2+25x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+25x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.12.12 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.12} & & = \frac{-25+\sqrt49}{2.12} \\ & = \frac{-32}{24} & & = \frac{-18}{24} \\ & = \frac{-4}{3} & & = \frac{-3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{-3}{4} \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{2}{3}x=-\frac{1}{15}x^2-\frac{5}{3} \\ \Leftrightarrow \frac{1}{15}x^2+\frac{2}{3}x+\frac{5}{3}=0 \\ \Leftrightarrow \color{red}{15.} \left(\frac{1}{15}x^2+\frac{2}{3}x+\frac{5}{3}\right)=0 \color{red}{.15} \\ \Leftrightarrow 4x^2+40x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+40x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (40)^2-4.4.100 & &\\ & = 1600-1600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-40}{2.4} & & \\ & = -5 & & \\V &= \Big\{ -5 \Big\} & &\end{align} \\ -----------------\)
  9. \(3x^2+\frac{7}{6}x-\frac{4}{3}=0\\ \Leftrightarrow \color{red}{6.} \left(3x^2+\frac{7}{6}x-\frac{4}{3}\right)=0 \color{red}{.6} \\ \Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  10. \(\frac{1}{5}x^2+\frac{5}{2}x+\frac{36}{5}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{1}{5}x^2+\frac{5}{2}x+\frac{36}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 2x^2+25x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+25x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.2.72 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.2} & & = \frac{-25+\sqrt49}{2.2} \\ & = \frac{-32}{4} & & = \frac{-18}{4} \\ & = -8 & & = \frac{-9}{2} \\ \\ V &= \Big\{ -8 ; \frac{-9}{2} \Big\} & &\end{align} \\ -----------------\)
  11. \(2x^2-(15x+32)=x(x-19) \\ \Leftrightarrow 2x^2-15x-32=x^2-19x \\ \Leftrightarrow x^2+4x-32=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-32=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-32) & &\\ & = 16+128 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt144}{2.1} & & = \frac{-4+\sqrt144}{2.1} \\ & = \frac{-16}{2} & & = \frac{8}{2} \\ & = -8 & & = 4 \\ \\ V &= \Big\{ -8 ; 4 \Big\} & &\end{align} \\ -----------------\)
  12. \(-(14+43x)=-9x^2-(95-11x) \\ \Leftrightarrow -14-43x=-9x^2-95+11x \\ \Leftrightarrow 9x^2-54x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-54x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-54)^2-4.9.81 & &\\ & = 2916-2916 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-54)}{2.9} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator vanhoeckes.be/wiskunde 2026-03-07 04:42:13