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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-(13-33x)=-24x^2-(19-8x)\)
  2. \(-(5-4x)=-9x^2-(9-16x)\)
  3. \(x(x-9)=3(x-12)\)
  4. \(3x^2-(12x+48)=2x(x-2)\)
  5. \(\frac{1}{5}x^2+\frac{5}{6}x+\frac{4}{5}=0\)
  6. \(-(13-9x)=-72x^2-(11-2x)\)
  7. \((5x+5)(-5x-3)-x(-61x-55)=-19\)
  8. \(x(8x+15)=-2(x+1)\)
  9. \(10x^2-(10x-100)=x(x+50)\)
  10. \(x(x-5)=4(x-5)\)
  11. \((x+2)(4x+2)-x(3x+2)=0\)
  12. \(\frac{25}{8}x=-\frac{9}{4}x^2-1\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-(13-33x)=-24x^2-(19-8x) \\ \Leftrightarrow -13+33x=-24x^2-19+8x \\ \Leftrightarrow 24x^2+25x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+25x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.24.6 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.24} & & = \frac{-25+\sqrt49}{2.24} \\ & = \frac{-32}{48} & & = \frac{-18}{48} \\ & = \frac{-2}{3} & & = \frac{-3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{-3}{8} \Big\} & &\end{align} \\ -----------------\)
  2. \(-(5-4x)=-9x^2-(9-16x) \\ \Leftrightarrow -5+4x=-9x^2-9+16x \\ \Leftrightarrow 9x^2-12x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-12x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.9.4 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-12)}{2.9} & & \\ & = \frac{2}{3} & & \\V &= \Big\{ \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
  3. \(x(x-9)=3(x-12) \\ \Leftrightarrow x^2-9x=3x-36 \\ \Leftrightarrow x^2-12x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.36 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-12)}{2.1} & & \\ & = 6 & & \\V &= \Big\{ 6 \Big\} & &\end{align} \\ -----------------\)
  4. \(3x^2-(12x+48)=2x(x-2) \\ \Leftrightarrow 3x^2-12x-48=2x^2-4x \\ \Leftrightarrow x^2-8x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-48) & &\\ & = 64+192 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt256}{2.1} & & = \frac{-(-8)+\sqrt256}{2.1} \\ & = \frac{-8}{2} & & = \frac{24}{2} \\ & = -4 & & = 12 \\ \\ V &= \Big\{ -4 ; 12 \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{1}{5}x^2+\frac{5}{6}x+\frac{4}{5}=0\\ \Leftrightarrow \color{red}{30.} \left(\frac{1}{5}x^2+\frac{5}{6}x+\frac{4}{5}\right)=0 \color{red}{.30} \\ \Leftrightarrow 6x^2+25x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+25x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.6.24 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.6} & & = \frac{-25+\sqrt49}{2.6} \\ & = \frac{-32}{12} & & = \frac{-18}{12} \\ & = \frac{-8}{3} & & = \frac{-3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{-3}{2} \Big\} & &\end{align} \\ -----------------\)
  6. \(-(13-9x)=-72x^2-(11-2x) \\ \Leftrightarrow -13+9x=-72x^2-11+2x \\ \Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.72.(-2) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\ & = \frac{-32}{144} & & = \frac{18}{144} \\ & = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  7. \((5x+5)(-5x-3)-x(-61x-55)=-19\\ \Leftrightarrow -25x^2-15x-25x-15 +61x^2+55x+19=0 \\ \Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  8. \(x(8x+15)=-2(x+1) \\ \Leftrightarrow 8x^2+15x=-2x-2 \\ \Leftrightarrow 8x^2+17x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+17x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.8.2 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.8} & & = \frac{-17+\sqrt225}{2.8} \\ & = \frac{-32}{16} & & = \frac{-2}{16} \\ & = -2 & & = \frac{-1}{8} \\ \\ V &= \Big\{ -2 ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
  9. \(10x^2-(10x-100)=x(x+50) \\ \Leftrightarrow 10x^2-10x+100=x^2+50x \\ \Leftrightarrow 9x^2-60x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-60x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-60)^2-4.9.100 & &\\ & = 3600-3600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-60)}{2.9} & & \\ & = \frac{10}{3} & & \\V &= \Big\{ \frac{10}{3} \Big\} & &\end{align} \\ -----------------\)
  10. \(x(x-5)=4(x-5) \\ \Leftrightarrow x^2-5x=4x-20 \\ \Leftrightarrow x^2-9x+20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x+20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-9)^2-4.1.20 & &\\ & = 81-80 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-9)-\sqrt1}{2.1} & & = \frac{-(-9)+\sqrt1}{2.1} \\ & = \frac{8}{2} & & = \frac{10}{2} \\ & = 4 & & = 5 \\ \\ V &= \Big\{ 4 ; 5 \Big\} & &\end{align} \\ -----------------\)
  11. \((x+2)(4x+2)-x(3x+2)=0\\ \Leftrightarrow 4x^2+2x+8x+4 -3x^2-2x+0=0 \\ \Leftrightarrow x^2+4x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.1} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
  12. \(\frac{25}{8}x=-\frac{9}{4}x^2-1 \\ \Leftrightarrow \frac{9}{4}x^2+\frac{25}{8}x+1=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{9}{4}x^2+\frac{25}{8}x+1\right)=0 \color{red}{.8} \\ \Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator vanhoeckes.be/wiskunde 2024-05-03 09:37:04