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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{1}{2}x^2+\frac{7}{2}x-9=0\)
  2. \((4x-3)(x+4)-x(3x+3)=18\)
  3. \(-\frac{7}{4}x=-\frac{1}{4}x^2+11\)
  4. \(x(4x-21)=-9(x+1)\)
  5. \(-(11+11x)=-x^2-(88-7x)\)
  6. \((-4x-4)(5x-2)-x(-29x-26)=-41\)
  7. \(x(16x+11)=3(x-3)\)
  8. \(2x^2-(2x+4)=x(x+1)\)
  9. \((5x-4)(-x-1)-x(-6x+17)=-76\)
  10. \(-(14+88x)=-16x^2-(158-8x)\)
  11. \(\frac{9}{5}x=-\frac{1}{5}x^2+\frac{22}{5}\)
  12. \(x(4x+43)=36(x+1)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{1}{2}x^2+\frac{7}{2}x-9=0\\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+\frac{7}{2}x-9\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.(-18) & &\\ & = 49+72 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt121}{2.1} & & = \frac{-7+\sqrt121}{2.1} \\ & = \frac{-18}{2} & & = \frac{4}{2} \\ & = -9 & & = 2 \\ \\ V &= \Big\{ -9 ; 2 \Big\} & &\end{align} \\ -----------------\)
  2. \((4x-3)(x+4)-x(3x+3)=18\\ \Leftrightarrow 4x^2+16x-3x-12 -3x^2-3x-18=0 \\ \Leftrightarrow x^2+x-30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-30) & &\\ & = 1+120 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt121}{2.1} & & = \frac{-1+\sqrt121}{2.1} \\ & = \frac{-12}{2} & & = \frac{10}{2} \\ & = -6 & & = 5 \\ \\ V &= \Big\{ -6 ; 5 \Big\} & &\end{align} \\ -----------------\)
  3. \(-\frac{7}{4}x=-\frac{1}{4}x^2+11 \\ \Leftrightarrow \frac{1}{4}x^2-\frac{7}{4}x-11=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{7}{4}x-11\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-7x-44=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x-44=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.(-44) & &\\ & = 49+176 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt225}{2.1} & & = \frac{-(-7)+\sqrt225}{2.1} \\ & = \frac{-8}{2} & & = \frac{22}{2} \\ & = -4 & & = 11 \\ \\ V &= \Big\{ -4 ; 11 \Big\} & &\end{align} \\ -----------------\)
  4. \(x(4x-21)=-9(x+1) \\ \Leftrightarrow 4x^2-21x=-9x-9 \\ \Leftrightarrow 4x^2-12x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-12x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.4.9 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-12)}{2.4} & & \\ & = \frac{3}{2} & & \\V &= \Big\{ \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
  5. \(-(11+11x)=-x^2-(88-7x) \\ \Leftrightarrow -11-11x=-x^2-88+7x \\ \Leftrightarrow x^2-18x+77=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-18x+77=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-18)^2-4.1.77 & &\\ & = 324-308 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-18)-\sqrt16}{2.1} & & = \frac{-(-18)+\sqrt16}{2.1} \\ & = \frac{14}{2} & & = \frac{22}{2} \\ & = 7 & & = 11 \\ \\ V &= \Big\{ 7 ; 11 \Big\} & &\end{align} \\ -----------------\)
  6. \((-4x-4)(5x-2)-x(-29x-26)=-41\\ \Leftrightarrow -20x^2+8x-20x+8 +29x^2+26x+41=0 \\ \Leftrightarrow 9x^2+42x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+42x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (42)^2-4.9.49 & &\\ & = 1764-1764 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-42}{2.9} & & \\ & = -\frac{7}{3} & & \\V &= \Big\{ -\frac{7}{3} \Big\} & &\end{align} \\ -----------------\)
  7. \(x(16x+11)=3(x-3) \\ \Leftrightarrow 16x^2+11x=3x-9 \\ \Leftrightarrow 16x^2+8x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+8x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.16.9 & &\\ & = 64-576 & & \\ & = -512 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  8. \(2x^2-(2x+4)=x(x+1) \\ \Leftrightarrow 2x^2-2x-4=x^2+x \\ \Leftrightarrow x^2-3x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-4) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt25}{2.1} & & = \frac{-(-3)+\sqrt25}{2.1} \\ & = \frac{-2}{2} & & = \frac{8}{2} \\ & = -1 & & = 4 \\ \\ V &= \Big\{ -1 ; 4 \Big\} & &\end{align} \\ -----------------\)
  9. \((5x-4)(-x-1)-x(-6x+17)=-76\\ \Leftrightarrow -5x^2-5x+4x+4 +6x^2-17x+76=0 \\ \Leftrightarrow x^2-18x+80=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-18x+80=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-18)^2-4.1.80 & &\\ & = 324-320 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-18)-\sqrt4}{2.1} & & = \frac{-(-18)+\sqrt4}{2.1} \\ & = \frac{16}{2} & & = \frac{20}{2} \\ & = 8 & & = 10 \\ \\ V &= \Big\{ 8 ; 10 \Big\} & &\end{align} \\ -----------------\)
  10. \(-(14+88x)=-16x^2-(158-8x) \\ \Leftrightarrow -14-88x=-16x^2-158+8x \\ \Leftrightarrow 16x^2-96x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-96x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-96)^2-4.16.144 & &\\ & = 9216-9216 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-96)}{2.16} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
  11. \(\frac{9}{5}x=-\frac{1}{5}x^2+\frac{22}{5} \\ \Leftrightarrow \frac{1}{5}x^2+\frac{9}{5}x-\frac{22}{5}=0 \\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+\frac{9}{5}x-\frac{22}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2+9x-22=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+9x-22=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (9)^2-4.1.(-22) & &\\ & = 81+88 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-9-\sqrt169}{2.1} & & = \frac{-9+\sqrt169}{2.1} \\ & = \frac{-22}{2} & & = \frac{4}{2} \\ & = -11 & & = 2 \\ \\ V &= \Big\{ -11 ; 2 \Big\} & &\end{align} \\ -----------------\)
  12. \(x(4x+43)=36(x+1) \\ \Leftrightarrow 4x^2+43x=36x+36 \\ \Leftrightarrow 4x^2+7x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+7x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.4.(-36) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.4} & & = \frac{-7+\sqrt625}{2.4} \\ & = \frac{-32}{8} & & = \frac{18}{8} \\ & = -4 & & = \frac{9}{4} \\ \\ V &= \Big\{ -4 ; \frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2025-11-22 20:20:25
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