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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-(2+27x)=-4x^2-(66-5x)\)
  2. \(2x^2-(19x+28)=x(x-16)\)
  3. \(-\frac{13}{2}x=-\frac{1}{2}x^2-15\)
  4. \(5x^2-(6x-100)=x(x-2)\)
  5. \(\frac{9}{100}x^2+\frac{2}{5}x+1=0\)
  6. \(x(36x+21)=-4(x+1)\)
  7. \(17x^2-(20x-9)=x(x-44)\)
  8. \(\frac{23}{2}x=-\frac{1}{2}x^2-66\)
  9. \(\frac{7}{15}x=-\frac{16}{5}x^2+\frac{1}{5}\)
  10. \(\frac{9}{10}x^2-3x+\frac{5}{2}=0\)
  11. \((x-1)(5x-5)-x(4x+1)=61\)
  12. \(-2x=-\frac{3}{7}x^2-\frac{7}{3}\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-(2+27x)=-4x^2-(66-5x) \\ \Leftrightarrow -2-27x=-4x^2-66+5x \\ \Leftrightarrow 4x^2-32x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-32x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-32)^2-4.4.64 & &\\ & = 1024-1024 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-32)}{2.4} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
  2. \(2x^2-(19x+28)=x(x-16) \\ \Leftrightarrow 2x^2-19x-28=x^2-16x \\ \Leftrightarrow x^2-3x-28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-28=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-28) & &\\ & = 9+112 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt121}{2.1} & & = \frac{-(-3)+\sqrt121}{2.1} \\ & = \frac{-8}{2} & & = \frac{14}{2} \\ & = -4 & & = 7 \\ \\ V &= \Big\{ -4 ; 7 \Big\} & &\end{align} \\ -----------------\)
  3. \(-\frac{13}{2}x=-\frac{1}{2}x^2-15 \\ \Leftrightarrow \frac{1}{2}x^2-\frac{13}{2}x+15=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-\frac{13}{2}x+15\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2-13x+30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-13x+30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-13)^2-4.1.30 & &\\ & = 169-120 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-13)-\sqrt49}{2.1} & & = \frac{-(-13)+\sqrt49}{2.1} \\ & = \frac{6}{2} & & = \frac{20}{2} \\ & = 3 & & = 10 \\ \\ V &= \Big\{ 3 ; 10 \Big\} & &\end{align} \\ -----------------\)
  4. \(5x^2-(6x-100)=x(x-2) \\ \Leftrightarrow 5x^2-6x+100=x^2-2x \\ \Leftrightarrow 4x^2-4x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-4x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.4.100 & &\\ & = 16-1600 & & \\ & = -1584 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  5. \(\frac{9}{100}x^2+\frac{2}{5}x+1=0\\ \Leftrightarrow \color{red}{100.} \left(\frac{9}{100}x^2+\frac{2}{5}x+1\right)=0 \color{red}{.100} \\ \Leftrightarrow 9x^2+40x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+40x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (40)^2-4.9.100 & &\\ & = 1600-3600 & & \\ & = -2000 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  6. \(x(36x+21)=-4(x+1) \\ \Leftrightarrow 36x^2+21x=-4x-4 \\ \Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  7. \(17x^2-(20x-9)=x(x-44) \\ \Leftrightarrow 17x^2-20x+9=x^2-44x \\ \Leftrightarrow 16x^2+24x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+24x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.16.9 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.16} & & \\ & = -\frac{3}{4} & & \\V &= \Big\{ -\frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{23}{2}x=-\frac{1}{2}x^2-66 \\ \Leftrightarrow \frac{1}{2}x^2+\frac{23}{2}x+66=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+\frac{23}{2}x+66\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2+23x+132=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+23x+132=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (23)^2-4.1.132 & &\\ & = 529-528 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-23-\sqrt1}{2.1} & & = \frac{-23+\sqrt1}{2.1} \\ & = \frac{-24}{2} & & = \frac{-22}{2} \\ & = -12 & & = -11 \\ \\ V &= \Big\{ -12 ; -11 \Big\} & &\end{align} \\ -----------------\)
  9. \(\frac{7}{15}x=-\frac{16}{5}x^2+\frac{1}{5} \\ \Leftrightarrow \frac{16}{5}x^2+\frac{7}{15}x-\frac{1}{5}=0 \\ \Leftrightarrow \color{red}{15.} \left(\frac{16}{5}x^2+\frac{7}{15}x-\frac{1}{5}\right)=0 \color{red}{.15} \\ \Leftrightarrow 48x^2+7x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
  10. \(\frac{9}{10}x^2-3x+\frac{5}{2}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{9}{10}x^2-3x+\frac{5}{2}\right)=0 \color{red}{.10} \\ \Leftrightarrow 9x^2-30x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-30x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-30)^2-4.9.25 & &\\ & = 900-900 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-30)}{2.9} & & \\ & = \frac{5}{3} & & \\V &= \Big\{ \frac{5}{3} \Big\} & &\end{align} \\ -----------------\)
  11. \((x-1)(5x-5)-x(4x+1)=61\\ \Leftrightarrow 5x^2-5x-5x+5 -4x^2-x-61=0 \\ \Leftrightarrow x^2-x-56=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-56=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-56) & &\\ & = 1+224 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt225}{2.1} & & = \frac{-(-1)+\sqrt225}{2.1} \\ & = \frac{-14}{2} & & = \frac{16}{2} \\ & = -7 & & = 8 \\ \\ V &= \Big\{ -7 ; 8 \Big\} & &\end{align} \\ -----------------\)
  12. \(-2x=-\frac{3}{7}x^2-\frac{7}{3} \\ \Leftrightarrow \frac{3}{7}x^2-2x+\frac{7}{3}=0 \\ \Leftrightarrow \color{red}{21.} \left(\frac{3}{7}x^2-2x+\frac{7}{3}\right)=0 \color{red}{.21} \\ \Leftrightarrow 9x^2-42x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-42x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-42)^2-4.9.49 & &\\ & = 1764-1764 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-42)}{2.9} & & \\ & = \frac{7}{3} & & \\V &= \Big\{ \frac{7}{3} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-04-20 09:10:49
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