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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-(13-16x)=-6x^2-(7-11x)\)
  2. \(-\frac{7}{3}x=-\frac{1}{6}x^2-8\)
  3. \(6x^2-(11x+72)=5x(x-2)\)
  4. \(-(13+3x)=-x^2-(38-7x)\)
  5. \(\frac{1}{6}x^2+\frac{5}{2}x+9=0\)
  6. \(x(4x-99)=-81(x+1)\)
  7. \(\frac{1}{3}x^2+x+\frac{2}{3}=0\)
  8. \(\frac{4}{5}x^2+\frac{5}{2}x+\frac{9}{5}=0\)
  9. \(4x^2-(13x+14)=3x(x-6)\)
  10. \((3x+2)(x-4)-x(x-45)=-80\)
  11. \(4x^2-(16x+48)=3x(x-8)\)
  12. \(-(15-17x)=-6x^2-(-9-10x)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-(13-16x)=-6x^2-(7-11x) \\ \Leftrightarrow -13+16x=-6x^2-7+11x \\ \Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
  2. \(-\frac{7}{3}x=-\frac{1}{6}x^2-8 \\ \Leftrightarrow \frac{1}{6}x^2-\frac{7}{3}x+8=0 \\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{6}x^2-\frac{7}{3}x+8\right)=0 \color{red}{.6} \\ \Leftrightarrow x^2-14x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.48 & &\\ & = 196-192 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-14)-\sqrt4}{2.1} & & = \frac{-(-14)+\sqrt4}{2.1} \\ & = \frac{12}{2} & & = \frac{16}{2} \\ & = 6 & & = 8 \\ \\ V &= \Big\{ 6 ; 8 \Big\} & &\end{align} \\ -----------------\)
  3. \(6x^2-(11x+72)=5x(x-2) \\ \Leftrightarrow 6x^2-11x-72=5x^2-10x \\ \Leftrightarrow x^2-x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-72) & &\\ & = 1+288 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt289}{2.1} & & = \frac{-(-1)+\sqrt289}{2.1} \\ & = \frac{-16}{2} & & = \frac{18}{2} \\ & = -8 & & = 9 \\ \\ V &= \Big\{ -8 ; 9 \Big\} & &\end{align} \\ -----------------\)
  4. \(-(13+3x)=-x^2-(38-7x) \\ \Leftrightarrow -13-3x=-x^2-38+7x \\ \Leftrightarrow x^2-10x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.25 & &\\ & = 100-100 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-10)}{2.1} & & \\ & = 5 & & \\V &= \Big\{ 5 \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{1}{6}x^2+\frac{5}{2}x+9=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{6}x^2+\frac{5}{2}x+9\right)=0 \color{red}{.6} \\ \Leftrightarrow x^2+15x+54=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+15x+54=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.1.54 & &\\ & = 225-216 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt9}{2.1} & & = \frac{-15+\sqrt9}{2.1} \\ & = \frac{-18}{2} & & = \frac{-12}{2} \\ & = -9 & & = -6 \\ \\ V &= \Big\{ -9 ; -6 \Big\} & &\end{align} \\ -----------------\)
  6. \(x(4x-99)=-81(x+1) \\ \Leftrightarrow 4x^2-99x=-81x-81 \\ \Leftrightarrow 4x^2-18x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-18x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-18)^2-4.4.81 & &\\ & = 324-1296 & & \\ & = -972 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  7. \(\frac{1}{3}x^2+x+\frac{2}{3}=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+x+\frac{2}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2+3x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.2 & &\\ & = 9-8 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt1}{2.1} & & = \frac{-3+\sqrt1}{2.1} \\ & = \frac{-4}{2} & & = \frac{-2}{2} \\ & = -2 & & = -1 \\ \\ V &= \Big\{ -2 ; -1 \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{4}{5}x^2+\frac{5}{2}x+\frac{9}{5}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{4}{5}x^2+\frac{5}{2}x+\frac{9}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 8x^2+25x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+25x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.8.18 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.8} & & = \frac{-25+\sqrt49}{2.8} \\ & = \frac{-32}{16} & & = \frac{-18}{16} \\ & = -2 & & = \frac{-9}{8} \\ \\ V &= \Big\{ -2 ; \frac{-9}{8} \Big\} & &\end{align} \\ -----------------\)
  9. \(4x^2-(13x+14)=3x(x-6) \\ \Leftrightarrow 4x^2-13x-14=3x^2-18x \\ \Leftrightarrow x^2+5x-14=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-14=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-14) & &\\ & = 25+56 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt81}{2.1} & & = \frac{-5+\sqrt81}{2.1} \\ & = \frac{-14}{2} & & = \frac{4}{2} \\ & = -7 & & = 2 \\ \\ V &= \Big\{ -7 ; 2 \Big\} & &\end{align} \\ -----------------\)
  10. \((3x+2)(x-4)-x(x-45)=-80\\ \Leftrightarrow 3x^2-12x+2x-8 -x^2+45x+80=0 \\ \Leftrightarrow 2x^2+25x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+25x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.2.72 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.2} & & = \frac{-25+\sqrt49}{2.2} \\ & = \frac{-32}{4} & & = \frac{-18}{4} \\ & = -8 & & = \frac{-9}{2} \\ \\ V &= \Big\{ -8 ; \frac{-9}{2} \Big\} & &\end{align} \\ -----------------\)
  11. \(4x^2-(16x+48)=3x(x-8) \\ \Leftrightarrow 4x^2-16x-48=3x^2-24x \\ \Leftrightarrow x^2+8x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.(-48) & &\\ & = 64+192 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-8-\sqrt256}{2.1} & & = \frac{-8+\sqrt256}{2.1} \\ & = \frac{-24}{2} & & = \frac{8}{2} \\ & = -12 & & = 4 \\ \\ V &= \Big\{ -12 ; 4 \Big\} & &\end{align} \\ -----------------\)
  12. \(-(15-17x)=-6x^2-(-9-10x) \\ \Leftrightarrow -15+17x=-6x^2+9+10x \\ \Leftrightarrow 6x^2+7x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+7x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.6.(-24) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.6} & & = \frac{-7+\sqrt625}{2.6} \\ & = \frac{-32}{12} & & = \frac{18}{12} \\ & = \frac{-8}{3} & & = \frac{3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-02-01 13:18:47
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