VKV met breuken of rekenwerk

Hoofdmenu Eentje per keer 

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-(7+2x)=-9x^2-(8-4x)\)
  2. \(\frac{7}{8}x=-9x^2+\frac{1}{4}\)
  3. \(x(x-9)=2(x-14)\)
  4. \(\frac{1}{2}x^2+\frac{5}{4}x+\frac{1}{2}=0\)
  5. \(\frac{4}{5}x^2+\frac{13}{5}x+\frac{9}{5}=0\)
  6. \(-(2+3x)=-x^2-(-34-6x)\)
  7. \(x(16x+16)=-(x+1)\)
  8. \((4x+2)(-3x-5)-x(-30x-43)=-12\)
  9. \(\frac{3}{5}x^2-\frac{8}{5}x+\frac{5}{3}=0\)
  10. \(x(9x-22)=2(x-8)\)
  11. \(x(8x+17)=2(x+1)\)
  12. \(\frac{1}{4}x^2+\frac{15}{16}x-\frac{1}{4}=0\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-(7+2x)=-9x^2-(8-4x) \\ \Leftrightarrow -7-2x=-9x^2-8+4x \\ \Leftrightarrow 9x^2-6x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-6x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.9.1 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.9} & & \\ & = \frac{1}{3} & & \\V &= \Big\{ \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
  2. \(\frac{7}{8}x=-9x^2+\frac{1}{4} \\ \Leftrightarrow 9x^2+\frac{7}{8}x-\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{8.} \left(9x^2+\frac{7}{8}x-\frac{1}{4}\right)=0 \color{red}{.8} \\ \Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.72.(-2) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\ & = \frac{-32}{144} & & = \frac{18}{144} \\ & = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  3. \(x(x-9)=2(x-14) \\ \Leftrightarrow x^2-9x=2x-28 \\ \Leftrightarrow x^2-11x+28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+28=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-11)^2-4.1.28 & &\\ & = 121-112 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-11)-\sqrt9}{2.1} & & = \frac{-(-11)+\sqrt9}{2.1} \\ & = \frac{8}{2} & & = \frac{14}{2} \\ & = 4 & & = 7 \\ \\ V &= \Big\{ 4 ; 7 \Big\} & &\end{align} \\ -----------------\)
  4. \(\frac{1}{2}x^2+\frac{5}{4}x+\frac{1}{2}=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{2}x^2+\frac{5}{4}x+\frac{1}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{4}{5}x^2+\frac{13}{5}x+\frac{9}{5}=0\\ \Leftrightarrow \color{red}{5.} \left(\frac{4}{5}x^2+\frac{13}{5}x+\frac{9}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow 4x^2+13x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+13x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.4.9 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.4} & & = \frac{-13+\sqrt25}{2.4} \\ & = \frac{-18}{8} & & = \frac{-8}{8} \\ & = \frac{-9}{4} & & = -1 \\ \\ V &= \Big\{ \frac{-9}{4} ; -1 \Big\} & &\end{align} \\ -----------------\)
  6. \(-(2+3x)=-x^2-(-34-6x) \\ \Leftrightarrow -2-3x=-x^2+34+6x \\ \Leftrightarrow x^2-9x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-9)^2-4.1.(-36) & &\\ & = 81+144 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-9)-\sqrt225}{2.1} & & = \frac{-(-9)+\sqrt225}{2.1} \\ & = \frac{-6}{2} & & = \frac{24}{2} \\ & = -3 & & = 12 \\ \\ V &= \Big\{ -3 ; 12 \Big\} & &\end{align} \\ -----------------\)
  7. \(x(16x+16)=-(x+1) \\ \Leftrightarrow 16x^2+16x=-x-1 \\ \Leftrightarrow 16x^2+17x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+17x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.16.1 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.16} & & = \frac{-17+\sqrt225}{2.16} \\ & = \frac{-32}{32} & & = \frac{-2}{32} \\ & = -1 & & = \frac{-1}{16} \\ \\ V &= \Big\{ -1 ; \frac{-1}{16} \Big\} & &\end{align} \\ -----------------\)
  8. \((4x+2)(-3x-5)-x(-30x-43)=-12\\ \Leftrightarrow -12x^2-20x-6x-10 +30x^2+43x+12=0 \\ \Leftrightarrow 18x^2+13x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+13x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.18.2 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.18} & & = \frac{-13+\sqrt25}{2.18} \\ & = \frac{-18}{36} & & = \frac{-8}{36} \\ & = \frac{-1}{2} & & = \frac{-2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-2}{9} \Big\} & &\end{align} \\ -----------------\)
  9. \(\frac{3}{5}x^2-\frac{8}{5}x+\frac{5}{3}=0\\ \Leftrightarrow \color{red}{15.} \left(\frac{3}{5}x^2-\frac{8}{5}x+\frac{5}{3}\right)=0 \color{red}{.15} \\ \Leftrightarrow 9x^2-24x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-24x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.9.25 & &\\ & = 576-900 & & \\ & = -324 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  10. \(x(9x-22)=2(x-8) \\ \Leftrightarrow 9x^2-22x=2x-16 \\ \Leftrightarrow 9x^2-24x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-24x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.9.16 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-24)}{2.9} & & \\ & = \frac{4}{3} & & \\V &= \Big\{ \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
  11. \(x(8x+17)=2(x+1) \\ \Leftrightarrow 8x^2+17x=2x+2 \\ \Leftrightarrow 8x^2+15x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+15x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.8.(-2) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.8} & & = \frac{-15+\sqrt289}{2.8} \\ & = \frac{-32}{16} & & = \frac{2}{16} \\ & = -2 & & = \frac{1}{8} \\ \\ V &= \Big\{ -2 ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  12. \(\frac{1}{4}x^2+\frac{15}{16}x-\frac{1}{4}=0\\ \Leftrightarrow \color{red}{16.} \left(\frac{1}{4}x^2+\frac{15}{16}x-\frac{1}{4}\right)=0 \color{red}{.16} \\ \Leftrightarrow 4x^2+15x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.4.(-4) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\ & = \frac{-32}{8} & & = \frac{2}{8} \\ & = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2025-04-02 07:05:55
Een site van Busleyden Atheneum Mechelen