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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-\frac{1}{3}x=-\frac{2}{3}x^2-\frac{1}{24}\)
  2. \(\frac{1}{4}x=-\frac{1}{16}x^2+\frac{15}{4}\)
  3. \((-x-1)(x-1)-x(-2x+17)=-49\)
  4. \(\frac{1}{6}x=-\frac{1}{5}x^2+\frac{1}{5}\)
  5. \(x(x+57)=54(x+1)\)
  6. \(2x^2-(4x+40)=x(x-7)\)
  7. \((-4x+5)(-2x+3)-x(-x+63)=-85\)
  8. \(-(12-16x)=-x^2-(36-5x)\)
  9. \(x(2x+5)=2(x+1)\)
  10. \(\frac{1}{2}x^2-6x+\frac{35}{2}=0\)
  11. \(x(x-53)=-49(x+1)\)
  12. \(9x^2-(12x+18)=x(x-19)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-\frac{1}{3}x=-\frac{2}{3}x^2-\frac{1}{24} \\ \Leftrightarrow \frac{2}{3}x^2-\frac{1}{3}x+\frac{1}{24}=0 \\ \Leftrightarrow \color{red}{24.} \left(\frac{2}{3}x^2-\frac{1}{3}x+\frac{1}{24}\right)=0 \color{red}{.24} \\ \Leftrightarrow 16x^2-8x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-8x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.16.1 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-8)}{2.16} & & \\ & = \frac{1}{4} & & \\V &= \Big\{ \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
  2. \(\frac{1}{4}x=-\frac{1}{16}x^2+\frac{15}{4} \\ \Leftrightarrow \frac{1}{16}x^2+\frac{1}{4}x-\frac{15}{4}=0 \\ \Leftrightarrow \color{red}{16.} \left(\frac{1}{16}x^2+\frac{1}{4}x-\frac{15}{4}\right)=0 \color{red}{.16} \\ \Leftrightarrow x^2+4x-60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-60) & &\\ & = 16+240 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt256}{2.1} & & = \frac{-4+\sqrt256}{2.1} \\ & = \frac{-20}{2} & & = \frac{12}{2} \\ & = -10 & & = 6 \\ \\ V &= \Big\{ -10 ; 6 \Big\} & &\end{align} \\ -----------------\)
  3. \((-x-1)(x-1)-x(-2x+17)=-49\\ \Leftrightarrow -x^2+x-x+1 +2x^2-17x+49=0 \\ \Leftrightarrow x^2-15x+50=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-15x+50=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-15)^2-4.1.50 & &\\ & = 225-200 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-15)-\sqrt25}{2.1} & & = \frac{-(-15)+\sqrt25}{2.1} \\ & = \frac{10}{2} & & = \frac{20}{2} \\ & = 5 & & = 10 \\ \\ V &= \Big\{ 5 ; 10 \Big\} & &\end{align} \\ -----------------\)
  4. \(\frac{1}{6}x=-\frac{1}{5}x^2+\frac{1}{5} \\ \Leftrightarrow \frac{1}{5}x^2+\frac{1}{6}x-\frac{1}{5}=0 \\ \Leftrightarrow \color{red}{30.} \left(\frac{1}{5}x^2+\frac{1}{6}x-\frac{1}{5}\right)=0 \color{red}{.30} \\ \Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
  5. \(x(x+57)=54(x+1) \\ \Leftrightarrow x^2+57x=54x+54 \\ \Leftrightarrow x^2+3x-54=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-54=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-54) & &\\ & = 9+216 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt225}{2.1} & & = \frac{-3+\sqrt225}{2.1} \\ & = \frac{-18}{2} & & = \frac{12}{2} \\ & = -9 & & = 6 \\ \\ V &= \Big\{ -9 ; 6 \Big\} & &\end{align} \\ -----------------\)
  6. \(2x^2-(4x+40)=x(x-7) \\ \Leftrightarrow 2x^2-4x-40=x^2-7x \\ \Leftrightarrow x^2+3x-40=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-40) & &\\ & = 9+160 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt169}{2.1} & & = \frac{-3+\sqrt169}{2.1} \\ & = \frac{-16}{2} & & = \frac{10}{2} \\ & = -8 & & = 5 \\ \\ V &= \Big\{ -8 ; 5 \Big\} & &\end{align} \\ -----------------\)
  7. \((-4x+5)(-2x+3)-x(-x+63)=-85\\ \Leftrightarrow 8x^2-12x-10x+15 +x^2-63x+85=0 \\ \Leftrightarrow 9x^2-60x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-60x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-60)^2-4.9.100 & &\\ & = 3600-3600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-60)}{2.9} & & \\ & = \frac{10}{3} & & \\V &= \Big\{ \frac{10}{3} \Big\} & &\end{align} \\ -----------------\)
  8. \(-(12-16x)=-x^2-(36-5x) \\ \Leftrightarrow -12+16x=-x^2-36+5x \\ \Leftrightarrow x^2+11x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.24 & &\\ & = 121-96 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt25}{2.1} & & = \frac{-11+\sqrt25}{2.1} \\ & = \frac{-16}{2} & & = \frac{-6}{2} \\ & = -8 & & = -3 \\ \\ V &= \Big\{ -8 ; -3 \Big\} & &\end{align} \\ -----------------\)
  9. \(x(2x+5)=2(x+1) \\ \Leftrightarrow 2x^2+5x=2x+2 \\ \Leftrightarrow 2x^2+3x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+3x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.2.(-2) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.2} & & = \frac{-3+\sqrt25}{2.2} \\ & = \frac{-8}{4} & & = \frac{2}{4} \\ & = -2 & & = \frac{1}{2} \\ \\ V &= \Big\{ -2 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  10. \(\frac{1}{2}x^2-6x+\frac{35}{2}=0\\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-6x+\frac{35}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2-12x+35=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+35=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.35 & &\\ & = 144-140 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-12)-\sqrt4}{2.1} & & = \frac{-(-12)+\sqrt4}{2.1} \\ & = \frac{10}{2} & & = \frac{14}{2} \\ & = 5 & & = 7 \\ \\ V &= \Big\{ 5 ; 7 \Big\} & &\end{align} \\ -----------------\)
  11. \(x(x-53)=-49(x+1) \\ \Leftrightarrow x^2-53x=-49x-49 \\ \Leftrightarrow x^2-4x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.49 & &\\ & = 16-196 & & \\ & = -180 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  12. \(9x^2-(12x+18)=x(x-19) \\ \Leftrightarrow 9x^2-12x-18=x^2-19x \\ \Leftrightarrow 8x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.8.(-18) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\ & = \frac{-32}{16} & & = \frac{18}{16} \\ & = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)
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