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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-\frac{1}{3}x=-\frac{1}{12}x^2+1\)
  2. \(x(x-5)=2(x-5)\)
  3. \(\frac{7}{6}x=-3x^2+\frac{4}{3}\)
  4. \(4x^2-(15x+7)=3x(x-3)\)
  5. \(\frac{9}{4}x^2+\frac{25}{8}x+1=0\)
  6. \(\frac{1}{3}x=-\frac{4}{5}x^2+\frac{1}{5}\)
  7. \(x(18x+11)=-2(x+1)\)
  8. \(\frac{1}{5}x^2-\frac{11}{5}x+6=0\)
  9. \(x(x+23)=8(x-7)\)
  10. \((5x-2)(-4x+4)-x(-92x+5)=-6\)
  11. \(-\frac{1}{4}x=-\frac{1}{4}x^2-\frac{1}{16}\)
  12. \(-(8+8x)=-x^2-(33-2x)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-\frac{1}{3}x=-\frac{1}{12}x^2+1 \\ \Leftrightarrow \frac{1}{12}x^2-\frac{1}{3}x-1=0 \\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{12}x^2-\frac{1}{3}x-1\right)=0 \color{red}{.12} \\ \Leftrightarrow x^2-4x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-12) & &\\ & = 16+48 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt64}{2.1} & & = \frac{-(-4)+\sqrt64}{2.1} \\ & = \frac{-4}{2} & & = \frac{12}{2} \\ & = -2 & & = 6 \\ \\ V &= \Big\{ -2 ; 6 \Big\} & &\end{align} \\ -----------------\)
  2. \(x(x-5)=2(x-5) \\ \Leftrightarrow x^2-5x=2x-10 \\ \Leftrightarrow x^2-7x+10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x+10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.10 & &\\ & = 49-40 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt9}{2.1} & & = \frac{-(-7)+\sqrt9}{2.1} \\ & = \frac{4}{2} & & = \frac{10}{2} \\ & = 2 & & = 5 \\ \\ V &= \Big\{ 2 ; 5 \Big\} & &\end{align} \\ -----------------\)
  3. \(\frac{7}{6}x=-3x^2+\frac{4}{3} \\ \Leftrightarrow 3x^2+\frac{7}{6}x-\frac{4}{3}=0 \\ \Leftrightarrow \color{red}{6.} \left(3x^2+\frac{7}{6}x-\frac{4}{3}\right)=0 \color{red}{.6} \\ \Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  4. \(4x^2-(15x+7)=3x(x-3) \\ \Leftrightarrow 4x^2-15x-7=3x^2-9x \\ \Leftrightarrow x^2-6x-7=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-7=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.(-7) & &\\ & = 36+28 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt64}{2.1} & & = \frac{-(-6)+\sqrt64}{2.1} \\ & = \frac{-2}{2} & & = \frac{14}{2} \\ & = -1 & & = 7 \\ \\ V &= \Big\{ -1 ; 7 \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{9}{4}x^2+\frac{25}{8}x+1=0\\ \Leftrightarrow \color{red}{8.} \left(\frac{9}{4}x^2+\frac{25}{8}x+1\right)=0 \color{red}{.8} \\ \Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
  6. \(\frac{1}{3}x=-\frac{4}{5}x^2+\frac{1}{5} \\ \Leftrightarrow \frac{4}{5}x^2+\frac{1}{3}x-\frac{1}{5}=0 \\ \Leftrightarrow \color{red}{15.} \left(\frac{4}{5}x^2+\frac{1}{3}x-\frac{1}{5}\right)=0 \color{red}{.15} \\ \Leftrightarrow 12x^2+5x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+5x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.12.(-3) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.12} & & = \frac{-5+\sqrt169}{2.12} \\ & = \frac{-18}{24} & & = \frac{8}{24} \\ & = \frac{-3}{4} & & = \frac{1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
  7. \(x(18x+11)=-2(x+1) \\ \Leftrightarrow 18x^2+11x=-2x-2 \\ \Leftrightarrow 18x^2+13x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+13x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.18.2 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.18} & & = \frac{-13+\sqrt25}{2.18} \\ & = \frac{-18}{36} & & = \frac{-8}{36} \\ & = \frac{-1}{2} & & = \frac{-2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-2}{9} \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{1}{5}x^2-\frac{11}{5}x+6=0\\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2-\frac{11}{5}x+6\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2-11x+30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-11)^2-4.1.30 & &\\ & = 121-120 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-11)-\sqrt1}{2.1} & & = \frac{-(-11)+\sqrt1}{2.1} \\ & = \frac{10}{2} & & = \frac{12}{2} \\ & = 5 & & = 6 \\ \\ V &= \Big\{ 5 ; 6 \Big\} & &\end{align} \\ -----------------\)
  9. \(x(x+23)=8(x-7) \\ \Leftrightarrow x^2+23x=8x-56 \\ \Leftrightarrow x^2+15x+56=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+15x+56=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.1.56 & &\\ & = 225-224 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt1}{2.1} & & = \frac{-15+\sqrt1}{2.1} \\ & = \frac{-16}{2} & & = \frac{-14}{2} \\ & = -8 & & = -7 \\ \\ V &= \Big\{ -8 ; -7 \Big\} & &\end{align} \\ -----------------\)
  10. \((5x-2)(-4x+4)-x(-92x+5)=-6\\ \Leftrightarrow -20x^2+20x+8x-8 +92x^2-5x+6=0 \\ \Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.72.(-2) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\ & = \frac{-32}{144} & & = \frac{18}{144} \\ & = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  11. \(-\frac{1}{4}x=-\frac{1}{4}x^2-\frac{1}{16} \\ \Leftrightarrow \frac{1}{4}x^2-\frac{1}{4}x+\frac{1}{16}=0 \\ \Leftrightarrow \color{red}{16.} \left(\frac{1}{4}x^2-\frac{1}{4}x+\frac{1}{16}\right)=0 \color{red}{.16} \\ \Leftrightarrow 4x^2-4x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-4x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.4.1 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-4)}{2.4} & & \\ & = \frac{1}{2} & & \\V &= \Big\{ \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  12. \(-(8+8x)=-x^2-(33-2x) \\ \Leftrightarrow -8-8x=-x^2-33+2x \\ \Leftrightarrow x^2-10x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.25 & &\\ & = 100-100 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-10)}{2.1} & & \\ & = 5 & & \\V &= \Big\{ 5 \Big\} & &\end{align} \\ -----------------\)
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