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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{7}{8}x=-\frac{9}{2}x^2+\frac{1}{2}\)
  2. \(-\frac{1}{4}x=-\frac{1}{8}x^2+6\)
  3. \(x(9x+9)=-4(x+1)\)
  4. \(x(x+36)=32(x+1)\)
  5. \(\frac{1}{20}x^2-\frac{4}{5}x+3=0\)
  6. \(10x^2-(13x-36)=x(x+23)\)
  7. \((x-3)(4x-5)-x(-12x+22)=-34\)
  8. \(-\frac{1}{5}x=-\frac{1}{60}x^2-\frac{3}{5}\)
  9. \((5x-1)(2x+2)-x(8x-5)=-20\)
  10. \(\frac{1}{3}x=-\frac{1}{6}x^2-\frac{2}{3}\)
  11. \(x(x+21)=20(x+1)\)
  12. \(-(4-28x)=-2x^2-(-4-13x)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{7}{8}x=-\frac{9}{2}x^2+\frac{1}{2} \\ \Leftrightarrow \frac{9}{2}x^2+\frac{7}{8}x-\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{9}{2}x^2+\frac{7}{8}x-\frac{1}{2}\right)=0 \color{red}{.8} \\ \Leftrightarrow 36x^2+7x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+7x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.36.(-4) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.36} & & = \frac{-7+\sqrt625}{2.36} \\ & = \frac{-32}{72} & & = \frac{18}{72} \\ & = \frac{-4}{9} & & = \frac{1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
  2. \(-\frac{1}{4}x=-\frac{1}{8}x^2+6 \\ \Leftrightarrow \frac{1}{8}x^2-\frac{1}{4}x-6=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2-\frac{1}{4}x-6\right)=0 \color{red}{.8} \\ \Leftrightarrow x^2-2x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-48) & &\\ & = 4+192 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt196}{2.1} & & = \frac{-(-2)+\sqrt196}{2.1} \\ & = \frac{-12}{2} & & = \frac{16}{2} \\ & = -6 & & = 8 \\ \\ V &= \Big\{ -6 ; 8 \Big\} & &\end{align} \\ -----------------\)
  3. \(x(9x+9)=-4(x+1) \\ \Leftrightarrow 9x^2+9x=-4x-4 \\ \Leftrightarrow 9x^2+13x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+13x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.9.4 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.9} & & = \frac{-13+\sqrt25}{2.9} \\ & = \frac{-18}{18} & & = \frac{-8}{18} \\ & = -1 & & = \frac{-4}{9} \\ \\ V &= \Big\{ -1 ; \frac{-4}{9} \Big\} & &\end{align} \\ -----------------\)
  4. \(x(x+36)=32(x+1) \\ \Leftrightarrow x^2+36x=32x+32 \\ \Leftrightarrow x^2+4x-32=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-32=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-32) & &\\ & = 16+128 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt144}{2.1} & & = \frac{-4+\sqrt144}{2.1} \\ & = \frac{-16}{2} & & = \frac{8}{2} \\ & = -8 & & = 4 \\ \\ V &= \Big\{ -8 ; 4 \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{1}{20}x^2-\frac{4}{5}x+3=0\\ \Leftrightarrow \color{red}{20.} \left(\frac{1}{20}x^2-\frac{4}{5}x+3\right)=0 \color{red}{.20} \\ \Leftrightarrow x^2-16x+60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.60 & &\\ & = 256-240 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-16)-\sqrt16}{2.1} & & = \frac{-(-16)+\sqrt16}{2.1} \\ & = \frac{12}{2} & & = \frac{20}{2} \\ & = 6 & & = 10 \\ \\ V &= \Big\{ 6 ; 10 \Big\} & &\end{align} \\ -----------------\)
  6. \(10x^2-(13x-36)=x(x+23) \\ \Leftrightarrow 10x^2-13x+36=x^2+23x \\ \Leftrightarrow 9x^2-36x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-36x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-36)^2-4.9.36 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-36)}{2.9} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
  7. \((x-3)(4x-5)-x(-12x+22)=-34\\ \Leftrightarrow 4x^2-5x-12x+15 +12x^2-22x+34=0 \\ \Leftrightarrow 16x^2-12x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-12x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.16.49 & &\\ & = 144-3136 & & \\ & = -2992 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  8. \(-\frac{1}{5}x=-\frac{1}{60}x^2-\frac{3}{5} \\ \Leftrightarrow \frac{1}{60}x^2-\frac{1}{5}x+\frac{3}{5}=0 \\ \Leftrightarrow \color{red}{60.} \left(\frac{1}{60}x^2-\frac{1}{5}x+\frac{3}{5}\right)=0 \color{red}{.60} \\ \Leftrightarrow x^2-12x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.36 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-12)}{2.1} & & \\ & = 6 & & \\V &= \Big\{ 6 \Big\} & &\end{align} \\ -----------------\)
  9. \((5x-1)(2x+2)-x(8x-5)=-20\\ \Leftrightarrow 10x^2+10x-2x-2 -8x^2+5x+20=0 \\ \Leftrightarrow 2x^2+13x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+13x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.2.18 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.2} & & = \frac{-13+\sqrt25}{2.2} \\ & = \frac{-18}{4} & & = \frac{-8}{4} \\ & = \frac{-9}{2} & & = -2 \\ \\ V &= \Big\{ \frac{-9}{2} ; -2 \Big\} & &\end{align} \\ -----------------\)
  10. \(\frac{1}{3}x=-\frac{1}{6}x^2-\frac{2}{3} \\ \Leftrightarrow \frac{1}{6}x^2+\frac{1}{3}x+\frac{2}{3}=0 \\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{6}x^2+\frac{1}{3}x+\frac{2}{3}\right)=0 \color{red}{.6} \\ \Leftrightarrow x^2+2x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.4 & &\\ & = 4-16 & & \\ & = -12 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  11. \(x(x+21)=20(x+1) \\ \Leftrightarrow x^2+21x=20x+20 \\ \Leftrightarrow x^2+x-20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-20) & &\\ & = 1+80 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt81}{2.1} & & = \frac{-1+\sqrt81}{2.1} \\ & = \frac{-10}{2} & & = \frac{8}{2} \\ & = -5 & & = 4 \\ \\ V &= \Big\{ -5 ; 4 \Big\} & &\end{align} \\ -----------------\)
  12. \(-(4-28x)=-2x^2-(-4-13x) \\ \Leftrightarrow -4+28x=-2x^2+4+13x \\ \Leftrightarrow 2x^2+15x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+15x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.2.(-8) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.2} & & = \frac{-15+\sqrt289}{2.2} \\ & = \frac{-32}{4} & & = \frac{2}{4} \\ & = -8 & & = \frac{1}{2} \\ \\ V &= \Big\{ -8 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-04-05 18:06:06
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