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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(x(x+23)=30(x+1)\)
  2. \(-x=-\frac{1}{22}x^2-\frac{11}{2}\)
  3. \(x(6x+11)=6(x+1)\)
  4. \(4x^2-(18x+12)=x(x-23)\)
  5. \(\frac{1}{5}x^2+\frac{1}{5}x-6=0\)
  6. \(5x^2-(13x+9)=x(x-18)\)
  7. \(-(9-25x)=-8x^2-(-9-18x)\)
  8. \(\frac{1}{3}x^2+\frac{7}{12}x-3=0\)
  9. \(x(x+20)=6(x-4)\)
  10. \((5x-1)(-3x+4)-x(-17x+3)=-22\)
  11. \(\frac{1}{8}x^2+\frac{1}{2}x+\frac{1}{2}=0\)
  12. \(-(6-15x)=-9x^2-(2-10x)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(x(x+23)=30(x+1) \\ \Leftrightarrow x^2+23x=30x+30 \\ \Leftrightarrow x^2-7x-30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x-30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.(-30) & &\\ & = 49+120 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt169}{2.1} & & = \frac{-(-7)+\sqrt169}{2.1} \\ & = \frac{-6}{2} & & = \frac{20}{2} \\ & = -3 & & = 10 \\ \\ V &= \Big\{ -3 ; 10 \Big\} & &\end{align} \\ -----------------\)
  2. \(-x=-\frac{1}{22}x^2-\frac{11}{2} \\ \Leftrightarrow \frac{1}{22}x^2-x+\frac{11}{2}=0 \\ \Leftrightarrow \color{red}{22.} \left(\frac{1}{22}x^2-x+\frac{11}{2}\right)=0 \color{red}{.22} \\ \Leftrightarrow x^2-22x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-22x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.1.121 & &\\ & = 484-484 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-22)}{2.1} & & \\ & = 11 & & \\V &= \Big\{ 11 \Big\} & &\end{align} \\ -----------------\)
  3. \(x(6x+11)=6(x+1) \\ \Leftrightarrow 6x^2+11x=6x+6 \\ \Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
  4. \(4x^2-(18x+12)=x(x-23) \\ \Leftrightarrow 4x^2-18x-12=x^2-23x \\ \Leftrightarrow 3x^2+5x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+5x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.3.(-12) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.3} & & = \frac{-5+\sqrt169}{2.3} \\ & = \frac{-18}{6} & & = \frac{8}{6} \\ & = -3 & & = \frac{4}{3} \\ \\ V &= \Big\{ -3 ; \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{1}{5}x^2+\frac{1}{5}x-6=0\\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+\frac{1}{5}x-6\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2+x-30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-30) & &\\ & = 1+120 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt121}{2.1} & & = \frac{-1+\sqrt121}{2.1} \\ & = \frac{-12}{2} & & = \frac{10}{2} \\ & = -6 & & = 5 \\ \\ V &= \Big\{ -6 ; 5 \Big\} & &\end{align} \\ -----------------\)
  6. \(5x^2-(13x+9)=x(x-18) \\ \Leftrightarrow 5x^2-13x-9=x^2-18x \\ \Leftrightarrow 4x^2+5x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.(-9) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.4} & & = \frac{-5+\sqrt169}{2.4} \\ & = \frac{-18}{8} & & = \frac{8}{8} \\ & = \frac{-9}{4} & & = 1 \\ \\ V &= \Big\{ \frac{-9}{4} ; 1 \Big\} & &\end{align} \\ -----------------\)
  7. \(-(9-25x)=-8x^2-(-9-18x) \\ \Leftrightarrow -9+25x=-8x^2+9+18x \\ \Leftrightarrow 8x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.8.(-18) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\ & = \frac{-32}{16} & & = \frac{18}{16} \\ & = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{1}{3}x^2+\frac{7}{12}x-3=0\\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{3}x^2+\frac{7}{12}x-3\right)=0 \color{red}{.12} \\ \Leftrightarrow 4x^2+7x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+7x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.4.(-36) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.4} & & = \frac{-7+\sqrt625}{2.4} \\ & = \frac{-32}{8} & & = \frac{18}{8} \\ & = -4 & & = \frac{9}{4} \\ \\ V &= \Big\{ -4 ; \frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
  9. \(x(x+20)=6(x-4) \\ \Leftrightarrow x^2+20x=6x-24 \\ \Leftrightarrow x^2+14x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.24 & &\\ & = 196-96 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-14-\sqrt100}{2.1} & & = \frac{-14+\sqrt100}{2.1} \\ & = \frac{-24}{2} & & = \frac{-4}{2} \\ & = -12 & & = -2 \\ \\ V &= \Big\{ -12 ; -2 \Big\} & &\end{align} \\ -----------------\)
  10. \((5x-1)(-3x+4)-x(-17x+3)=-22\\ \Leftrightarrow -15x^2+20x+3x-4 +17x^2-3x+22=0 \\ \Leftrightarrow 2x^2+13x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+13x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.2.18 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.2} & & = \frac{-13+\sqrt25}{2.2} \\ & = \frac{-18}{4} & & = \frac{-8}{4} \\ & = \frac{-9}{2} & & = -2 \\ \\ V &= \Big\{ \frac{-9}{2} ; -2 \Big\} & &\end{align} \\ -----------------\)
  11. \(\frac{1}{8}x^2+\frac{1}{2}x+\frac{1}{2}=0\\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2+\frac{1}{2}x+\frac{1}{2}\right)=0 \color{red}{.8} \\ \Leftrightarrow 9x^2+36x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+36x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (36)^2-4.9.36 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-36}{2.9} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
  12. \(-(6-15x)=-9x^2-(2-10x) \\ \Leftrightarrow -6+15x=-9x^2-2+10x \\ \Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.9.(-4) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\ & = \frac{-18}{18} & & = \frac{8}{18} \\ & = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-07-20 17:11:40
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