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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{9}{2}x^2-x+\frac{121}{2}=0\)
  2. \(\frac{1}{4}x^2+\frac{5}{12}x-1=0\)
  3. \(x(9x-55)=-25(x+1)\)
  4. \((3x+5)(-2x-4)-x(-7x-38)=7\)
  5. \((2x-4)(-2x+3)-x(-22x-19)=-14\)
  6. \(x(36x+11)=4(x+1)\)
  7. \(x(18x+15)=8(x+1)\)
  8. \(37x^2-(6x-4)=x(x-31)\)
  9. \(\frac{6}{5}x=-\frac{9}{25}x^2-1\)
  10. \(-\frac{1}{5}x=-\frac{1}{5}x^2+\frac{2}{5}\)
  11. \(2x^2-(8x+27)=x(x-2)\)
  12. \(x(x+11)=6(x+1)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{9}{2}x^2-x+\frac{121}{2}=0\\ \Leftrightarrow \color{red}{2.} \left(\frac{9}{2}x^2-x+\frac{121}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow 9x^2-2x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-2x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.9.121 & &\\ & = 4-4356 & & \\ & = -4352 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  2. \(\frac{1}{4}x^2+\frac{5}{12}x-1=0\\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{4}x^2+\frac{5}{12}x-1\right)=0 \color{red}{.12} \\ \Leftrightarrow 3x^2+5x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+5x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.3.(-12) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.3} & & = \frac{-5+\sqrt169}{2.3} \\ & = \frac{-18}{6} & & = \frac{8}{6} \\ & = -3 & & = \frac{4}{3} \\ \\ V &= \Big\{ -3 ; \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
  3. \(x(9x-55)=-25(x+1) \\ \Leftrightarrow 9x^2-55x=-25x-25 \\ \Leftrightarrow 9x^2-30x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-30x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-30)^2-4.9.25 & &\\ & = 900-900 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-30)}{2.9} & & \\ & = \frac{5}{3} & & \\V &= \Big\{ \frac{5}{3} \Big\} & &\end{align} \\ -----------------\)
  4. \((3x+5)(-2x-4)-x(-7x-38)=7\\ \Leftrightarrow -6x^2-12x-10x-20 +7x^2+38x-7=0 \\ \Leftrightarrow x^2+6x-27=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x-27=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.(-27) & &\\ & = 36+108 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt144}{2.1} & & = \frac{-6+\sqrt144}{2.1} \\ & = \frac{-18}{2} & & = \frac{6}{2} \\ & = -9 & & = 3 \\ \\ V &= \Big\{ -9 ; 3 \Big\} & &\end{align} \\ -----------------\)
  5. \((2x-4)(-2x+3)-x(-22x-19)=-14\\ \Leftrightarrow -4x^2+6x+8x-12 +22x^2+19x+14=0 \\ \Leftrightarrow 18x^2+13x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+13x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.18.2 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.18} & & = \frac{-13+\sqrt25}{2.18} \\ & = \frac{-18}{36} & & = \frac{-8}{36} \\ & = \frac{-1}{2} & & = \frac{-2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-2}{9} \Big\} & &\end{align} \\ -----------------\)
  6. \(x(36x+11)=4(x+1) \\ \Leftrightarrow 36x^2+11x=4x+4 \\ \Leftrightarrow 36x^2+7x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+7x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.36.(-4) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.36} & & = \frac{-7+\sqrt625}{2.36} \\ & = \frac{-32}{72} & & = \frac{18}{72} \\ & = \frac{-4}{9} & & = \frac{1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
  7. \(x(18x+15)=8(x+1) \\ \Leftrightarrow 18x^2+15x=8x+8 \\ \Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  8. \(37x^2-(6x-4)=x(x-31) \\ \Leftrightarrow 37x^2-6x+4=x^2-31x \\ \Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  9. \(\frac{6}{5}x=-\frac{9}{25}x^2-1 \\ \Leftrightarrow \frac{9}{25}x^2+\frac{6}{5}x+1=0 \\ \Leftrightarrow \color{red}{25.} \left(\frac{9}{25}x^2+\frac{6}{5}x+1\right)=0 \color{red}{.25} \\ \Leftrightarrow 9x^2+30x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+30x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (30)^2-4.9.25 & &\\ & = 900-900 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-30}{2.9} & & \\ & = -\frac{5}{3} & & \\V &= \Big\{ -\frac{5}{3} \Big\} & &\end{align} \\ -----------------\)
  10. \(-\frac{1}{5}x=-\frac{1}{5}x^2+\frac{2}{5} \\ \Leftrightarrow \frac{1}{5}x^2-\frac{1}{5}x-\frac{2}{5}=0 \\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2-\frac{1}{5}x-\frac{2}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2-x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-2) & &\\ & = 1+8 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt9}{2.1} & & = \frac{-(-1)+\sqrt9}{2.1} \\ & = \frac{-2}{2} & & = \frac{4}{2} \\ & = -1 & & = 2 \\ \\ V &= \Big\{ -1 ; 2 \Big\} & &\end{align} \\ -----------------\)
  11. \(2x^2-(8x+27)=x(x-2) \\ \Leftrightarrow 2x^2-8x-27=x^2-2x \\ \Leftrightarrow x^2-6x-27=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-27=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.(-27) & &\\ & = 36+108 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt144}{2.1} & & = \frac{-(-6)+\sqrt144}{2.1} \\ & = \frac{-6}{2} & & = \frac{18}{2} \\ & = -3 & & = 9 \\ \\ V &= \Big\{ -3 ; 9 \Big\} & &\end{align} \\ -----------------\)
  12. \(x(x+11)=6(x+1) \\ \Leftrightarrow x^2+11x=6x+6 \\ \Leftrightarrow x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-6) & &\\ & = 25+24 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt49}{2.1} & & = \frac{-5+\sqrt49}{2.1} \\ & = \frac{-12}{2} & & = \frac{2}{2} \\ & = -6 & & = 1 \\ \\ V &= \Big\{ -6 ; 1 \Big\} & &\end{align} \\ -----------------\)
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