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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-\frac{1}{3}x=-\frac{1}{3}x^2-\frac{1}{12}\)
  2. \(7x^2-(15x-48)=4x(x-10)\)
  3. \(x(x+5)=3(x+1)\)
  4. \(x(18x+7)=2(x+1)\)
  5. \(-\frac{3}{4}x=-\frac{9}{16}x^2-\frac{1}{4}\)
  6. \((4x-2)(5x-4)-x(16x-40)=-56\)
  7. \(x(x+7)=4(x+1)\)
  8. \((-4x+2)(-5x+4)-x(18x-33)=-64\)
  9. \(53x^2-(10x-3)=5x(x-7)\)
  10. \((3x+4)(4x-2)-x(8x+30)=-129\)
  11. \((-x-1)(x+4)-x(-17x+34)=-53\)
  12. \(\frac{1}{45}x^2-\frac{1}{5}x+\frac{2}{5}=0\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-\frac{1}{3}x=-\frac{1}{3}x^2-\frac{1}{12} \\ \Leftrightarrow \frac{1}{3}x^2-\frac{1}{3}x+\frac{1}{12}=0 \\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{3}x^2-\frac{1}{3}x+\frac{1}{12}\right)=0 \color{red}{.12} \\ \Leftrightarrow 4x^2-4x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-4x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.4.1 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-4)}{2.4} & & \\ & = \frac{1}{2} & & \\V &= \Big\{ \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  2. \(7x^2-(15x-48)=4x(x-10) \\ \Leftrightarrow 7x^2-15x+48=4x^2-40x \\ \Leftrightarrow 3x^2+25x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+25x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.3.48 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.3} & & = \frac{-25+\sqrt49}{2.3} \\ & = \frac{-32}{6} & & = \frac{-18}{6} \\ & = \frac{-16}{3} & & = -3 \\ \\ V &= \Big\{ \frac{-16}{3} ; -3 \Big\} & &\end{align} \\ -----------------\)
  3. \(x(x+5)=3(x+1) \\ \Leftrightarrow x^2+5x=3x+3 \\ \Leftrightarrow x^2+2x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-3) & &\\ & = 4+12 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt16}{2.1} & & = \frac{-2+\sqrt16}{2.1} \\ & = \frac{-6}{2} & & = \frac{2}{2} \\ & = -3 & & = 1 \\ \\ V &= \Big\{ -3 ; 1 \Big\} & &\end{align} \\ -----------------\)
  4. \(x(18x+7)=2(x+1) \\ \Leftrightarrow 18x^2+7x=2x+2 \\ \Leftrightarrow 18x^2+5x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+5x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.18.(-2) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.18} & & = \frac{-5+\sqrt169}{2.18} \\ & = \frac{-18}{36} & & = \frac{8}{36} \\ & = \frac{-1}{2} & & = \frac{2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{2}{9} \Big\} & &\end{align} \\ -----------------\)
  5. \(-\frac{3}{4}x=-\frac{9}{16}x^2-\frac{1}{4} \\ \Leftrightarrow \frac{9}{16}x^2-\frac{3}{4}x+\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{16.} \left(\frac{9}{16}x^2-\frac{3}{4}x+\frac{1}{4}\right)=0 \color{red}{.16} \\ \Leftrightarrow 9x^2-12x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-12x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.9.4 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-12)}{2.9} & & \\ & = \frac{2}{3} & & \\V &= \Big\{ \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
  6. \((4x-2)(5x-4)-x(16x-40)=-56\\ \Leftrightarrow 20x^2-16x-10x+8 -16x^2+40x+56=0 \\ \Leftrightarrow 4x^2+32x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+32x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (32)^2-4.4.64 & &\\ & = 1024-1024 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-32}{2.4} & & \\ & = -4 & & \\V &= \Big\{ -4 \Big\} & &\end{align} \\ -----------------\)
  7. \(x(x+7)=4(x+1) \\ \Leftrightarrow x^2+7x=4x+4 \\ \Leftrightarrow x^2+3x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-4) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.1} & & = \frac{-3+\sqrt25}{2.1} \\ & = \frac{-8}{2} & & = \frac{2}{2} \\ & = -4 & & = 1 \\ \\ V &= \Big\{ -4 ; 1 \Big\} & &\end{align} \\ -----------------\)
  8. \((-4x+2)(-5x+4)-x(18x-33)=-64\\ \Leftrightarrow 20x^2-16x-10x+8 -18x^2+33x+64=0 \\ \Leftrightarrow 2x^2+25x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+25x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.2.72 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.2} & & = \frac{-25+\sqrt49}{2.2} \\ & = \frac{-32}{4} & & = \frac{-18}{4} \\ & = -8 & & = \frac{-9}{2} \\ \\ V &= \Big\{ -8 ; \frac{-9}{2} \Big\} & &\end{align} \\ -----------------\)
  9. \(53x^2-(10x-3)=5x(x-7) \\ \Leftrightarrow 53x^2-10x+3=5x^2-35x \\ \Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.48.3 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\ & = \frac{-32}{96} & & = \frac{-18}{96} \\ & = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
  10. \((3x+4)(4x-2)-x(8x+30)=-129\\ \Leftrightarrow 12x^2-6x+16x-8 -8x^2-30x+129=0 \\ \Leftrightarrow 4x^2-44x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-44x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-44)^2-4.4.121 & &\\ & = 1936-1936 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-44)}{2.4} & & \\ & = \frac{11}{2} & & \\V &= \Big\{ \frac{11}{2} \Big\} & &\end{align} \\ -----------------\)
  11. \((-x-1)(x+4)-x(-17x+34)=-53\\ \Leftrightarrow -x^2-4x-x-4 +17x^2-34x+53=0 \\ \Leftrightarrow 16x^2-42x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-42x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-42)^2-4.16.49 & &\\ & = 1764-3136 & & \\ & = -1372 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  12. \(\frac{1}{45}x^2-\frac{1}{5}x+\frac{2}{5}=0\\ \Leftrightarrow \color{red}{45.} \left(\frac{1}{45}x^2-\frac{1}{5}x+\frac{2}{5}\right)=0 \color{red}{.45} \\ \Leftrightarrow x^2-9x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-9)^2-4.1.18 & &\\ & = 81-72 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-9)-\sqrt9}{2.1} & & = \frac{-(-9)+\sqrt9}{2.1} \\ & = \frac{6}{2} & & = \frac{12}{2} \\ & = 3 & & = 6 \\ \\ V &= \Big\{ 3 ; 6 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator vanhoeckes.be/wiskunde 2025-05-19 10:44:40