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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-\frac{1}{3}x=-\frac{1}{12}x^2+8\)
  2. \(\frac{1}{6}x^2+\frac{5}{3}x-4=0\)
  3. \(2x^2-(3x-33)=x(x+11)\)
  4. \(\frac{1}{3}x^2+\frac{25}{12}x+3=0\)
  5. \(\frac{1}{2}x^2+6x+\frac{35}{2}=0\)
  6. \(\frac{3}{5}x=-\frac{1}{5}x^2-\frac{9}{20}\)
  7. \(\frac{1}{2}x^2+\frac{7}{8}x-\frac{9}{2}=0\)
  8. \(9x^2-(3x-72)=7x(x-4)\)
  9. \(x(x+33)=35(x+1)\)
  10. \((5x+2)(-3x-5)-x(-16x-46)=2\)
  11. \((5x-4)(-5x-4)-x(-26x+5)=-2\)
  12. \(2x^2-(10x-4)=x(x-14)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-\frac{1}{3}x=-\frac{1}{12}x^2+8 \\ \Leftrightarrow \frac{1}{12}x^2-\frac{1}{3}x-8=0 \\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{12}x^2-\frac{1}{3}x-8\right)=0 \color{red}{.12} \\ \Leftrightarrow x^2-4x-96=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-96=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-96) & &\\ & = 16+384 & & \\ & = 400 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt400}{2.1} & & = \frac{-(-4)+\sqrt400}{2.1} \\ & = \frac{-16}{2} & & = \frac{24}{2} \\ & = -8 & & = 12 \\ \\ V &= \Big\{ -8 ; 12 \Big\} & &\end{align} \\ -----------------\)
  2. \(\frac{1}{6}x^2+\frac{5}{3}x-4=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{6}x^2+\frac{5}{3}x-4\right)=0 \color{red}{.6} \\ \Leftrightarrow x^2+10x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+10x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.1.(-24) & &\\ & = 100+96 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt196}{2.1} & & = \frac{-10+\sqrt196}{2.1} \\ & = \frac{-24}{2} & & = \frac{4}{2} \\ & = -12 & & = 2 \\ \\ V &= \Big\{ -12 ; 2 \Big\} & &\end{align} \\ -----------------\)
  3. \(2x^2-(3x-33)=x(x+11) \\ \Leftrightarrow 2x^2-3x+33=x^2+11x \\ \Leftrightarrow x^2-14x+33=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+33=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.33 & &\\ & = 196-132 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-14)-\sqrt64}{2.1} & & = \frac{-(-14)+\sqrt64}{2.1} \\ & = \frac{6}{2} & & = \frac{22}{2} \\ & = 3 & & = 11 \\ \\ V &= \Big\{ 3 ; 11 \Big\} & &\end{align} \\ -----------------\)
  4. \(\frac{1}{3}x^2+\frac{25}{12}x+3=0\\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{3}x^2+\frac{25}{12}x+3\right)=0 \color{red}{.12} \\ \Leftrightarrow 4x^2+25x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+25x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.4.36 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.4} & & = \frac{-25+\sqrt49}{2.4} \\ & = \frac{-32}{8} & & = \frac{-18}{8} \\ & = -4 & & = \frac{-9}{4} \\ \\ V &= \Big\{ -4 ; \frac{-9}{4} \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{1}{2}x^2+6x+\frac{35}{2}=0\\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+6x+\frac{35}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2+12x+35=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+35=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.35 & &\\ & = 144-140 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-12-\sqrt4}{2.1} & & = \frac{-12+\sqrt4}{2.1} \\ & = \frac{-14}{2} & & = \frac{-10}{2} \\ & = -7 & & = -5 \\ \\ V &= \Big\{ -7 ; -5 \Big\} & &\end{align} \\ -----------------\)
  6. \(\frac{3}{5}x=-\frac{1}{5}x^2-\frac{9}{20} \\ \Leftrightarrow \frac{1}{5}x^2+\frac{3}{5}x+\frac{9}{20}=0 \\ \Leftrightarrow \color{red}{20.} \left(\frac{1}{5}x^2+\frac{3}{5}x+\frac{9}{20}\right)=0 \color{red}{.20} \\ \Leftrightarrow 4x^2+12x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+12x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.4.9 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-12}{2.4} & & \\ & = -\frac{3}{2} & & \\V &= \Big\{ -\frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
  7. \(\frac{1}{2}x^2+\frac{7}{8}x-\frac{9}{2}=0\\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{2}x^2+\frac{7}{8}x-\frac{9}{2}\right)=0 \color{red}{.8} \\ \Leftrightarrow 4x^2+7x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+7x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.4.(-36) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.4} & & = \frac{-7+\sqrt625}{2.4} \\ & = \frac{-32}{8} & & = \frac{18}{8} \\ & = -4 & & = \frac{9}{4} \\ \\ V &= \Big\{ -4 ; \frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
  8. \(9x^2-(3x-72)=7x(x-4) \\ \Leftrightarrow 9x^2-3x+72=7x^2-28x \\ \Leftrightarrow 2x^2+25x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+25x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.2.72 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.2} & & = \frac{-25+\sqrt49}{2.2} \\ & = \frac{-32}{4} & & = \frac{-18}{4} \\ & = -8 & & = \frac{-9}{2} \\ \\ V &= \Big\{ -8 ; \frac{-9}{2} \Big\} & &\end{align} \\ -----------------\)
  9. \(x(x+33)=35(x+1) \\ \Leftrightarrow x^2+33x=35x+35 \\ \Leftrightarrow x^2-2x-35=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-35=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-35) & &\\ & = 4+140 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt144}{2.1} & & = \frac{-(-2)+\sqrt144}{2.1} \\ & = \frac{-10}{2} & & = \frac{14}{2} \\ & = -5 & & = 7 \\ \\ V &= \Big\{ -5 ; 7 \Big\} & &\end{align} \\ -----------------\)
  10. \((5x+2)(-3x-5)-x(-16x-46)=2\\ \Leftrightarrow -15x^2-25x-6x-10 +16x^2+46x-2=0 \\ \Leftrightarrow x^2+11x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.(-12) & &\\ & = 121+48 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt169}{2.1} & & = \frac{-11+\sqrt169}{2.1} \\ & = \frac{-24}{2} & & = \frac{2}{2} \\ & = -12 & & = 1 \\ \\ V &= \Big\{ -12 ; 1 \Big\} & &\end{align} \\ -----------------\)
  11. \((5x-4)(-5x-4)-x(-26x+5)=-2\\ \Leftrightarrow -25x^2-20x+20x+16 +26x^2-5x+2=0 \\ \Leftrightarrow x^2-9x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-9)^2-4.1.18 & &\\ & = 81-72 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-9)-\sqrt9}{2.1} & & = \frac{-(-9)+\sqrt9}{2.1} \\ & = \frac{6}{2} & & = \frac{12}{2} \\ & = 3 & & = 6 \\ \\ V &= \Big\{ 3 ; 6 \Big\} & &\end{align} \\ -----------------\)
  12. \(2x^2-(10x-4)=x(x-14) \\ \Leftrightarrow 2x^2-10x+4=x^2-14x \\ \Leftrightarrow x^2+4x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.1} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
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