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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{1}{12}x^2+\frac{1}{3}x+\frac{1}{3}=0\)
  2. \(x(16x-69)=3(x-27)\)
  3. \((-3x-2)(x-2)-x(-19x+18)=-5\)
  4. \(-(10-8x)=-2x^2-(12-3x)\)
  5. \((-3x-4)(-x+5)-x(2x-24)=-38\)
  6. \(-(10+5x)=-x^2-(74-11x)\)
  7. \(-(8-12x)=-x^2-(-91-14x)\)
  8. \(-\frac{1}{5}x=-\frac{1}{45}x^2+\frac{4}{5}\)
  9. \(\frac{17}{8}x=-x^2-\frac{1}{4}\)
  10. \((-5x-2)(-3x+3)-x(-x-27)=-5\)
  11. \(8x^2-(3x-24)=2x(x-14)\)
  12. \(-(13-49x)=-4x^2-(134-5x)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{1}{12}x^2+\frac{1}{3}x+\frac{1}{3}=0\\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{12}x^2+\frac{1}{3}x+\frac{1}{3}\right)=0 \color{red}{.12} \\ \Leftrightarrow 9x^2+36x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+36x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (36)^2-4.9.36 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-36}{2.9} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
  2. \(x(16x-69)=3(x-27) \\ \Leftrightarrow 16x^2-69x=3x-81 \\ \Leftrightarrow 16x^2-72x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-72x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-72)^2-4.16.81 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-72)}{2.16} & & \\ & = \frac{9}{4} & & \\V &= \Big\{ \frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
  3. \((-3x-2)(x-2)-x(-19x+18)=-5\\ \Leftrightarrow -3x^2+6x-2x+4 +19x^2-18x+5=0 \\ \Leftrightarrow 16x^2-8x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-8x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.16.9 & &\\ & = 64-576 & & \\ & = -512 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  4. \(-(10-8x)=-2x^2-(12-3x) \\ \Leftrightarrow -10+8x=-2x^2-12+3x \\ \Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
  5. \((-3x-4)(-x+5)-x(2x-24)=-38\\ \Leftrightarrow 3x^2-15x+4x-20 -2x^2+24x+38=0 \\ \Leftrightarrow x^2-11x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-11)^2-4.1.18 & &\\ & = 121-72 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-11)-\sqrt49}{2.1} & & = \frac{-(-11)+\sqrt49}{2.1} \\ & = \frac{4}{2} & & = \frac{18}{2} \\ & = 2 & & = 9 \\ \\ V &= \Big\{ 2 ; 9 \Big\} & &\end{align} \\ -----------------\)
  6. \(-(10+5x)=-x^2-(74-11x) \\ \Leftrightarrow -10-5x=-x^2-74+11x \\ \Leftrightarrow x^2-16x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.64 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.1} & & \\ & = 8 & & \\V &= \Big\{ 8 \Big\} & &\end{align} \\ -----------------\)
  7. \(-(8-12x)=-x^2-(-91-14x) \\ \Leftrightarrow -8+12x=-x^2+91+14x \\ \Leftrightarrow x^2-2x-99=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-99=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-99) & &\\ & = 4+396 & & \\ & = 400 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt400}{2.1} & & = \frac{-(-2)+\sqrt400}{2.1} \\ & = \frac{-18}{2} & & = \frac{22}{2} \\ & = -9 & & = 11 \\ \\ V &= \Big\{ -9 ; 11 \Big\} & &\end{align} \\ -----------------\)
  8. \(-\frac{1}{5}x=-\frac{1}{45}x^2+\frac{4}{5} \\ \Leftrightarrow \frac{1}{45}x^2-\frac{1}{5}x-\frac{4}{5}=0 \\ \Leftrightarrow \color{red}{45.} \left(\frac{1}{45}x^2-\frac{1}{5}x-\frac{4}{5}\right)=0 \color{red}{.45} \\ \Leftrightarrow x^2-9x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-9)^2-4.1.(-36) & &\\ & = 81+144 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-9)-\sqrt225}{2.1} & & = \frac{-(-9)+\sqrt225}{2.1} \\ & = \frac{-6}{2} & & = \frac{24}{2} \\ & = -3 & & = 12 \\ \\ V &= \Big\{ -3 ; 12 \Big\} & &\end{align} \\ -----------------\)
  9. \(\frac{17}{8}x=-x^2-\frac{1}{4} \\ \Leftrightarrow x^2+\frac{17}{8}x+\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{8.} \left(x^2+\frac{17}{8}x+\frac{1}{4}\right)=0 \color{red}{.8} \\ \Leftrightarrow 8x^2+17x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+17x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.8.2 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.8} & & = \frac{-17+\sqrt225}{2.8} \\ & = \frac{-32}{16} & & = \frac{-2}{16} \\ & = -2 & & = \frac{-1}{8} \\ \\ V &= \Big\{ -2 ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
  10. \((-5x-2)(-3x+3)-x(-x-27)=-5\\ \Leftrightarrow 15x^2-15x+6x-6 +x^2+27x+5=0 \\ \Leftrightarrow 16x^2+6x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+6x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.16.(-1) & &\\ & = 36+64 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt100}{2.16} & & = \frac{-6+\sqrt100}{2.16} \\ & = \frac{-16}{32} & & = \frac{4}{32} \\ & = \frac{-1}{2} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  11. \(8x^2-(3x-24)=2x(x-14) \\ \Leftrightarrow 8x^2-3x+24=2x^2-28x \\ \Leftrightarrow 6x^2+25x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+25x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.6.24 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.6} & & = \frac{-25+\sqrt49}{2.6} \\ & = \frac{-32}{12} & & = \frac{-18}{12} \\ & = \frac{-8}{3} & & = \frac{-3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{-3}{2} \Big\} & &\end{align} \\ -----------------\)
  12. \(-(13-49x)=-4x^2-(134-5x) \\ \Leftrightarrow -13+49x=-4x^2-134+5x \\ \Leftrightarrow 4x^2+44x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+44x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (44)^2-4.4.121 & &\\ & = 1936-1936 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-44}{2.4} & & \\ & = -\frac{11}{2} & & \\V &= \Big\{ -\frac{11}{2} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2025-11-30 21:06:40
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