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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(x(9x-20)=2(x-32)\)
  2. \((-3x-3)(-5x+1)-x(13x-31)=-75\)
  3. \(x(x+8)=12(x+1)\)
  4. \(7x^2-(13x-4)=3x(x-10)\)
  5. \(-(4-26x)=-18x^2-(-4-19x)\)
  6. \(-(13-8x)=-x^2-(-50-6x)\)
  7. \(-(13-33x)=-4x^2-(49-9x)\)
  8. \(\frac{17}{6}x=-\frac{1}{3}x^2-\frac{4}{3}\)
  9. \(x(72x+23)=-2(x+1)\)
  10. \(x(x+10)=8(x+1)\)
  11. \(\frac{1}{2}x=-\frac{1}{5}x^2-\frac{1}{5}\)
  12. \(-(5-20x)=-36x^2-(1-13x)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(x(9x-20)=2(x-32) \\ \Leftrightarrow 9x^2-20x=2x-64 \\ \Leftrightarrow 9x^2-22x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-22x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.9.64 & &\\ & = 484-2304 & & \\ & = -1820 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  2. \((-3x-3)(-5x+1)-x(13x-31)=-75\\ \Leftrightarrow 15x^2-3x+15x-3 -13x^2+31x+75=0 \\ \Leftrightarrow 2x^2+25x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+25x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.2.72 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.2} & & = \frac{-25+\sqrt49}{2.2} \\ & = \frac{-32}{4} & & = \frac{-18}{4} \\ & = -8 & & = \frac{-9}{2} \\ \\ V &= \Big\{ -8 ; \frac{-9}{2} \Big\} & &\end{align} \\ -----------------\)
  3. \(x(x+8)=12(x+1) \\ \Leftrightarrow x^2+8x=12x+12 \\ \Leftrightarrow x^2-4x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-12) & &\\ & = 16+48 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt64}{2.1} & & = \frac{-(-4)+\sqrt64}{2.1} \\ & = \frac{-4}{2} & & = \frac{12}{2} \\ & = -2 & & = 6 \\ \\ V &= \Big\{ -2 ; 6 \Big\} & &\end{align} \\ -----------------\)
  4. \(7x^2-(13x-4)=3x(x-10) \\ \Leftrightarrow 7x^2-13x+4=3x^2-30x \\ \Leftrightarrow 4x^2+17x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+17x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.4.4 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.4} & & = \frac{-17+\sqrt225}{2.4} \\ & = \frac{-32}{8} & & = \frac{-2}{8} \\ & = -4 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -4 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  5. \(-(4-26x)=-18x^2-(-4-19x) \\ \Leftrightarrow -4+26x=-18x^2+4+19x \\ \Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  6. \(-(13-8x)=-x^2-(-50-6x) \\ \Leftrightarrow -13+8x=-x^2+50+6x \\ \Leftrightarrow x^2+2x-63=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-63=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-63) & &\\ & = 4+252 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt256}{2.1} & & = \frac{-2+\sqrt256}{2.1} \\ & = \frac{-18}{2} & & = \frac{14}{2} \\ & = -9 & & = 7 \\ \\ V &= \Big\{ -9 ; 7 \Big\} & &\end{align} \\ -----------------\)
  7. \(-(13-33x)=-4x^2-(49-9x) \\ \Leftrightarrow -13+33x=-4x^2-49+9x \\ \Leftrightarrow 4x^2+24x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+24x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.4.36 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.4} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{17}{6}x=-\frac{1}{3}x^2-\frac{4}{3} \\ \Leftrightarrow \frac{1}{3}x^2+\frac{17}{6}x+\frac{4}{3}=0 \\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{3}x^2+\frac{17}{6}x+\frac{4}{3}\right)=0 \color{red}{.6} \\ \Leftrightarrow 2x^2+17x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+17x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.2.8 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.2} & & = \frac{-17+\sqrt225}{2.2} \\ & = \frac{-32}{4} & & = \frac{-2}{4} \\ & = -8 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -8 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
  9. \(x(72x+23)=-2(x+1) \\ \Leftrightarrow 72x^2+23x=-2x-2 \\ \Leftrightarrow 72x^2+25x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+25x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.72.2 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.72} & & = \frac{-25+\sqrt49}{2.72} \\ & = \frac{-32}{144} & & = \frac{-18}{144} \\ & = \frac{-2}{9} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
  10. \(x(x+10)=8(x+1) \\ \Leftrightarrow x^2+10x=8x+8 \\ \Leftrightarrow x^2+2x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-8) & &\\ & = 4+32 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt36}{2.1} & & = \frac{-2+\sqrt36}{2.1} \\ & = \frac{-8}{2} & & = \frac{4}{2} \\ & = -4 & & = 2 \\ \\ V &= \Big\{ -4 ; 2 \Big\} & &\end{align} \\ -----------------\)
  11. \(\frac{1}{2}x=-\frac{1}{5}x^2-\frac{1}{5} \\ \Leftrightarrow \frac{1}{5}x^2+\frac{1}{2}x+\frac{1}{5}=0 \\ \Leftrightarrow \color{red}{10.} \left(\frac{1}{5}x^2+\frac{1}{2}x+\frac{1}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
  12. \(-(5-20x)=-36x^2-(1-13x) \\ \Leftrightarrow -5+20x=-36x^2-1+13x \\ \Leftrightarrow 36x^2+7x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+7x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.36.(-4) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.36} & & = \frac{-7+\sqrt625}{2.36} \\ & = \frac{-32}{72} & & = \frac{18}{72} \\ & = \frac{-4}{9} & & = \frac{1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2025-04-03 23:32:17
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