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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(17x^2-(4x+1)=x(x-19)\)
  2. \(\frac{17}{2}x=-\frac{1}{2}x^2-35\)
  3. \(x(24x+27)=2(x-3)\)
  4. \(58x^2-(13x+3)=10x(x-2)\)
  5. \(-(14-61x)=-16x^2-(158-15x)\)
  6. \(x(4x-77)=-49(x+1)\)
  7. \((-5x-1)(3x-5)-x(-19x+10)=-20\)
  8. \(\frac{5}{8}x=-\frac{1}{4}x^2-\frac{1}{4}\)
  9. \(\frac{1}{6}x^2-x+\frac{3}{2}=0\)
  10. \(\frac{25}{8}x=-\frac{1}{4}x^2-9\)
  11. \(\frac{1}{4}x^2+\frac{5}{12}x-1=0\)
  12. \(2x^2-(19x+9)=x(x-11)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(17x^2-(4x+1)=x(x-19) \\ \Leftrightarrow 17x^2-4x-1=x^2-19x \\ \Leftrightarrow 16x^2+15x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+15x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.16.(-1) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.16} & & = \frac{-15+\sqrt289}{2.16} \\ & = \frac{-32}{32} & & = \frac{2}{32} \\ & = -1 & & = \frac{1}{16} \\ \\ V &= \Big\{ -1 ; \frac{1}{16} \Big\} & &\end{align} \\ -----------------\)
  2. \(\frac{17}{2}x=-\frac{1}{2}x^2-35 \\ \Leftrightarrow \frac{1}{2}x^2+\frac{17}{2}x+35=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+\frac{17}{2}x+35\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2+17x+70=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+17x+70=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.1.70 & &\\ & = 289-280 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt9}{2.1} & & = \frac{-17+\sqrt9}{2.1} \\ & = \frac{-20}{2} & & = \frac{-14}{2} \\ & = -10 & & = -7 \\ \\ V &= \Big\{ -10 ; -7 \Big\} & &\end{align} \\ -----------------\)
  3. \(x(24x+27)=2(x-3) \\ \Leftrightarrow 24x^2+27x=2x-6 \\ \Leftrightarrow 24x^2+25x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+25x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.24.6 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.24} & & = \frac{-25+\sqrt49}{2.24} \\ & = \frac{-32}{48} & & = \frac{-18}{48} \\ & = \frac{-2}{3} & & = \frac{-3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{-3}{8} \Big\} & &\end{align} \\ -----------------\)
  4. \(58x^2-(13x+3)=10x(x-2) \\ \Leftrightarrow 58x^2-13x-3=10x^2-20x \\ \Leftrightarrow 48x^2+7x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
  5. \(-(14-61x)=-16x^2-(158-15x) \\ \Leftrightarrow -14+61x=-16x^2-158+15x \\ \Leftrightarrow 16x^2+46x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+46x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (46)^2-4.16.144 & &\\ & = 2116-9216 & & \\ & = -7100 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  6. \(x(4x-77)=-49(x+1) \\ \Leftrightarrow 4x^2-77x=-49x-49 \\ \Leftrightarrow 4x^2-28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-28x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-28)^2-4.4.49 & &\\ & = 784-784 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-28)}{2.4} & & \\ & = \frac{7}{2} & & \\V &= \Big\{ \frac{7}{2} \Big\} & &\end{align} \\ -----------------\)
  7. \((-5x-1)(3x-5)-x(-19x+10)=-20\\ \Leftrightarrow -15x^2+25x-3x+5 +19x^2-10x+20=0 \\ \Leftrightarrow 4x^2+20x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+20x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.4.25 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.4} & & \\ & = -\frac{5}{2} & & \\V &= \Big\{ -\frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{5}{8}x=-\frac{1}{4}x^2-\frac{1}{4} \\ \Leftrightarrow \frac{1}{4}x^2+\frac{5}{8}x+\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{4}x^2+\frac{5}{8}x+\frac{1}{4}\right)=0 \color{red}{.8} \\ \Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
  9. \(\frac{1}{6}x^2-x+\frac{3}{2}=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{6}x^2-x+\frac{3}{2}\right)=0 \color{red}{.6} \\ \Leftrightarrow x^2-6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.9 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.1} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
  10. \(\frac{25}{8}x=-\frac{1}{4}x^2-9 \\ \Leftrightarrow \frac{1}{4}x^2+\frac{25}{8}x+9=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{4}x^2+\frac{25}{8}x+9\right)=0 \color{red}{.8} \\ \Leftrightarrow 2x^2+25x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+25x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.2.72 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.2} & & = \frac{-25+\sqrt49}{2.2} \\ & = \frac{-32}{4} & & = \frac{-18}{4} \\ & = -8 & & = \frac{-9}{2} \\ \\ V &= \Big\{ -8 ; \frac{-9}{2} \Big\} & &\end{align} \\ -----------------\)
  11. \(\frac{1}{4}x^2+\frac{5}{12}x-1=0\\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{4}x^2+\frac{5}{12}x-1\right)=0 \color{red}{.12} \\ \Leftrightarrow 3x^2+5x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+5x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.3.(-12) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.3} & & = \frac{-5+\sqrt169}{2.3} \\ & = \frac{-18}{6} & & = \frac{8}{6} \\ & = -3 & & = \frac{4}{3} \\ \\ V &= \Big\{ -3 ; \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
  12. \(2x^2-(19x+9)=x(x-11) \\ \Leftrightarrow 2x^2-19x-9=x^2-11x \\ \Leftrightarrow x^2-8x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-9) & &\\ & = 64+36 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt100}{2.1} & & = \frac{-(-8)+\sqrt100}{2.1} \\ & = \frac{-2}{2} & & = \frac{18}{2} \\ & = -1 & & = 9 \\ \\ V &= \Big\{ -1 ; 9 \Big\} & &\end{align} \\ -----------------\)
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