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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(9x^2-(16x+6)=3x(x-7)\)
  2. \(-(14-10x)=-16x^2-(15-18x)\)
  3. \(x(x+91)=88(x+1)\)
  4. \(x(x+61)=66(x+1)\)
  5. \(-(8-x)=-x^2-(-42-6x)\)
  6. \(\frac{4}{5}x^2+\frac{7}{10}x-\frac{9}{5}=0\)
  7. \(\frac{1}{33}x^2-\frac{2}{3}x+\frac{11}{3}=0\)
  8. \(-(4-38x)=-24x^2-(10-13x)\)
  9. \(8x^2-(12x+22)=7x(x-3)\)
  10. \(-(13-32x)=-2x^2-(21-15x)\)
  11. \(x(3x+16)=3(x-4)\)
  12. \(\frac{1}{2}x^2-2x-\frac{21}{2}=0\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(9x^2-(16x+6)=3x(x-7) \\ \Leftrightarrow 9x^2-16x-6=3x^2-21x \\ \Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
  2. \(-(14-10x)=-16x^2-(15-18x) \\ \Leftrightarrow -14+10x=-16x^2-15+18x \\ \Leftrightarrow 16x^2-8x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-8x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.16.1 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-8)}{2.16} & & \\ & = \frac{1}{4} & & \\V &= \Big\{ \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
  3. \(x(x+91)=88(x+1) \\ \Leftrightarrow x^2+91x=88x+88 \\ \Leftrightarrow x^2+3x-88=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-88=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-88) & &\\ & = 9+352 & & \\ & = 361 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt361}{2.1} & & = \frac{-3+\sqrt361}{2.1} \\ & = \frac{-22}{2} & & = \frac{16}{2} \\ & = -11 & & = 8 \\ \\ V &= \Big\{ -11 ; 8 \Big\} & &\end{align} \\ -----------------\)
  4. \(x(x+61)=66(x+1) \\ \Leftrightarrow x^2+61x=66x+66 \\ \Leftrightarrow x^2-5x-66=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-66=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-66) & &\\ & = 25+264 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt289}{2.1} & & = \frac{-(-5)+\sqrt289}{2.1} \\ & = \frac{-12}{2} & & = \frac{22}{2} \\ & = -6 & & = 11 \\ \\ V &= \Big\{ -6 ; 11 \Big\} & &\end{align} \\ -----------------\)
  5. \(-(8-x)=-x^2-(-42-6x) \\ \Leftrightarrow -8+x=-x^2+42+6x \\ \Leftrightarrow x^2-5x-50=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-50=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-50) & &\\ & = 25+200 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt225}{2.1} & & = \frac{-(-5)+\sqrt225}{2.1} \\ & = \frac{-10}{2} & & = \frac{20}{2} \\ & = -5 & & = 10 \\ \\ V &= \Big\{ -5 ; 10 \Big\} & &\end{align} \\ -----------------\)
  6. \(\frac{4}{5}x^2+\frac{7}{10}x-\frac{9}{5}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{4}{5}x^2+\frac{7}{10}x-\frac{9}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 8x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.8.(-18) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\ & = \frac{-32}{16} & & = \frac{18}{16} \\ & = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)
  7. \(\frac{1}{33}x^2-\frac{2}{3}x+\frac{11}{3}=0\\ \Leftrightarrow \color{red}{33.} \left(\frac{1}{33}x^2-\frac{2}{3}x+\frac{11}{3}\right)=0 \color{red}{.33} \\ \Leftrightarrow x^2-22x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-22x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.1.121 & &\\ & = 484-484 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-22)}{2.1} & & \\ & = 11 & & \\V &= \Big\{ 11 \Big\} & &\end{align} \\ -----------------\)
  8. \(-(4-38x)=-24x^2-(10-13x) \\ \Leftrightarrow -4+38x=-24x^2-10+13x \\ \Leftrightarrow 24x^2+25x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+25x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.24.6 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.24} & & = \frac{-25+\sqrt49}{2.24} \\ & = \frac{-32}{48} & & = \frac{-18}{48} \\ & = \frac{-2}{3} & & = \frac{-3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{-3}{8} \Big\} & &\end{align} \\ -----------------\)
  9. \(8x^2-(12x+22)=7x(x-3) \\ \Leftrightarrow 8x^2-12x-22=7x^2-21x \\ \Leftrightarrow x^2+9x-22=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+9x-22=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (9)^2-4.1.(-22) & &\\ & = 81+88 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-9-\sqrt169}{2.1} & & = \frac{-9+\sqrt169}{2.1} \\ & = \frac{-22}{2} & & = \frac{4}{2} \\ & = -11 & & = 2 \\ \\ V &= \Big\{ -11 ; 2 \Big\} & &\end{align} \\ -----------------\)
  10. \(-(13-32x)=-2x^2-(21-15x) \\ \Leftrightarrow -13+32x=-2x^2-21+15x \\ \Leftrightarrow 2x^2+17x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+17x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.2.8 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.2} & & = \frac{-17+\sqrt225}{2.2} \\ & = \frac{-32}{4} & & = \frac{-2}{4} \\ & = -8 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -8 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
  11. \(x(3x+16)=3(x-4) \\ \Leftrightarrow 3x^2+16x=3x-12 \\ \Leftrightarrow 3x^2+13x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+13x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.3.12 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.3} & & = \frac{-13+\sqrt25}{2.3} \\ & = \frac{-18}{6} & & = \frac{-8}{6} \\ & = -3 & & = \frac{-4}{3} \\ \\ V &= \Big\{ -3 ; \frac{-4}{3} \Big\} & &\end{align} \\ -----------------\)
  12. \(\frac{1}{2}x^2-2x-\frac{21}{2}=0\\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-2x-\frac{21}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2-4x-21=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-21=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-21) & &\\ & = 16+84 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt100}{2.1} & & = \frac{-(-4)+\sqrt100}{2.1} \\ & = \frac{-6}{2} & & = \frac{14}{2} \\ & = -3 & & = 7 \\ \\ V &= \Big\{ -3 ; 7 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-08-24 22:53:53
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