VKV met breuken of rekenwerk

Hoofdmenu Eentje per keer 

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \((2x-1)(-5x-5)-x(-14x-25)=-20\)
  2. \(-(15-19x)=-9x^2-(51-13x)\)
  3. \(2x^2-(18x-28)=x(x-7)\)
  4. \(-\frac{7}{5}x=-\frac{1}{5}x^2-\frac{6}{5}\)
  5. \(-(3-x)=-x^2-(-4-7x)\)
  6. \(-\frac{1}{2}x=-\frac{1}{4}x^2+\frac{3}{4}\)
  7. \(3x^2-(15x+30)=2x(x-8)\)
  8. \(-(13-17x)=-18x^2-(15-4x)\)
  9. \(2x^2-(16x-132)=x(x+7)\)
  10. \(-(12-18x)=-x^2-(10-17x)\)
  11. \(\frac{1}{3}x^2+\frac{5}{2}x-\frac{4}{3}=0\)
  12. \(\frac{17}{4}x=-2x^2-\frac{1}{2}\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \((2x-1)(-5x-5)-x(-14x-25)=-20\\ \Leftrightarrow -10x^2-10x+5x+5 +14x^2+25x+20=0 \\ \Leftrightarrow 4x^2+20x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+20x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.4.25 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.4} & & \\ & = -\frac{5}{2} & & \\V &= \Big\{ -\frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
  2. \(-(15-19x)=-9x^2-(51-13x) \\ \Leftrightarrow -15+19x=-9x^2-51+13x \\ \Leftrightarrow 9x^2+6x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+6x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.9.36 & &\\ & = 36-1296 & & \\ & = -1260 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  3. \(2x^2-(18x-28)=x(x-7) \\ \Leftrightarrow 2x^2-18x+28=x^2-7x \\ \Leftrightarrow x^2-11x+28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+28=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-11)^2-4.1.28 & &\\ & = 121-112 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-11)-\sqrt9}{2.1} & & = \frac{-(-11)+\sqrt9}{2.1} \\ & = \frac{8}{2} & & = \frac{14}{2} \\ & = 4 & & = 7 \\ \\ V &= \Big\{ 4 ; 7 \Big\} & &\end{align} \\ -----------------\)
  4. \(-\frac{7}{5}x=-\frac{1}{5}x^2-\frac{6}{5} \\ \Leftrightarrow \frac{1}{5}x^2-\frac{7}{5}x+\frac{6}{5}=0 \\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2-\frac{7}{5}x+\frac{6}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2-7x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.6 & &\\ & = 49-24 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt25}{2.1} & & = \frac{-(-7)+\sqrt25}{2.1} \\ & = \frac{2}{2} & & = \frac{12}{2} \\ & = 1 & & = 6 \\ \\ V &= \Big\{ 1 ; 6 \Big\} & &\end{align} \\ -----------------\)
  5. \(-(3-x)=-x^2-(-4-7x) \\ \Leftrightarrow -3+x=-x^2+4+7x \\ \Leftrightarrow x^2-6x-7=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-7=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.(-7) & &\\ & = 36+28 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt64}{2.1} & & = \frac{-(-6)+\sqrt64}{2.1} \\ & = \frac{-2}{2} & & = \frac{14}{2} \\ & = -1 & & = 7 \\ \\ V &= \Big\{ -1 ; 7 \Big\} & &\end{align} \\ -----------------\)
  6. \(-\frac{1}{2}x=-\frac{1}{4}x^2+\frac{3}{4} \\ \Leftrightarrow \frac{1}{4}x^2-\frac{1}{2}x-\frac{3}{4}=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{1}{2}x-\frac{3}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-2x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-3) & &\\ & = 4+12 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt16}{2.1} & & = \frac{-(-2)+\sqrt16}{2.1} \\ & = \frac{-2}{2} & & = \frac{6}{2} \\ & = -1 & & = 3 \\ \\ V &= \Big\{ -1 ; 3 \Big\} & &\end{align} \\ -----------------\)
  7. \(3x^2-(15x+30)=2x(x-8) \\ \Leftrightarrow 3x^2-15x-30=2x^2-16x \\ \Leftrightarrow x^2+x-30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-30) & &\\ & = 1+120 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt121}{2.1} & & = \frac{-1+\sqrt121}{2.1} \\ & = \frac{-12}{2} & & = \frac{10}{2} \\ & = -6 & & = 5 \\ \\ V &= \Big\{ -6 ; 5 \Big\} & &\end{align} \\ -----------------\)
  8. \(-(13-17x)=-18x^2-(15-4x) \\ \Leftrightarrow -13+17x=-18x^2-15+4x \\ \Leftrightarrow 18x^2+13x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+13x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.18.2 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.18} & & = \frac{-13+\sqrt25}{2.18} \\ & = \frac{-18}{36} & & = \frac{-8}{36} \\ & = \frac{-1}{2} & & = \frac{-2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-2}{9} \Big\} & &\end{align} \\ -----------------\)
  9. \(2x^2-(16x-132)=x(x+7) \\ \Leftrightarrow 2x^2-16x+132=x^2+7x \\ \Leftrightarrow x^2-23x+132=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-23x+132=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-23)^2-4.1.132 & &\\ & = 529-528 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-23)-\sqrt1}{2.1} & & = \frac{-(-23)+\sqrt1}{2.1} \\ & = \frac{22}{2} & & = \frac{24}{2} \\ & = 11 & & = 12 \\ \\ V &= \Big\{ 11 ; 12 \Big\} & &\end{align} \\ -----------------\)
  10. \(-(12-18x)=-x^2-(10-17x) \\ \Leftrightarrow -12+18x=-x^2-10+17x \\ \Leftrightarrow x^2+x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-2) & &\\ & = 1+8 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt9}{2.1} & & = \frac{-1+\sqrt9}{2.1} \\ & = \frac{-4}{2} & & = \frac{2}{2} \\ & = -2 & & = 1 \\ \\ V &= \Big\{ -2 ; 1 \Big\} & &\end{align} \\ -----------------\)
  11. \(\frac{1}{3}x^2+\frac{5}{2}x-\frac{4}{3}=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{3}x^2+\frac{5}{2}x-\frac{4}{3}\right)=0 \color{red}{.6} \\ \Leftrightarrow 2x^2+15x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+15x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.2.(-8) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.2} & & = \frac{-15+\sqrt289}{2.2} \\ & = \frac{-32}{4} & & = \frac{2}{4} \\ & = -8 & & = \frac{1}{2} \\ \\ V &= \Big\{ -8 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  12. \(\frac{17}{4}x=-2x^2-\frac{1}{2} \\ \Leftrightarrow 2x^2+\frac{17}{4}x+\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{4.} \left(2x^2+\frac{17}{4}x+\frac{1}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow 8x^2+17x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+17x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.8.2 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.8} & & = \frac{-17+\sqrt225}{2.8} \\ & = \frac{-32}{16} & & = \frac{-2}{16} \\ & = -2 & & = \frac{-1}{8} \\ \\ V &= \Big\{ -2 ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-10-06 10:57:18
Een site van Busleyden Atheneum Mechelen