VKV met breuken of rekenwerk

Hoofdmenu Eentje per keer 

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \((-5x+5)(4x+4)-x(-24x-13)=11\)
  2. \(3x^2-(20x-24)=2x(x-5)\)
  3. \(-\frac{1}{2}x=-\frac{1}{4}x^2-4\)
  4. \(-(5-9x)=-48x^2-(2-2x)\)
  5. \(-(2-44x)=-8x^2-(20-19x)\)
  6. \(\frac{16}{9}x^2+\frac{2}{3}x+9=0\)
  7. \(\frac{4}{3}x^2+\frac{5}{3}x-3=0\)
  8. \(x=-\frac{1}{3}x^2-\frac{4}{3}\)
  9. \(12x^2+\frac{7}{6}x-\frac{1}{3}=0\)
  10. \(-(14-8x)=-x^2-(-21-6x)\)
  11. \(x(x-120)=-100(x+1)\)
  12. \(8x^2-(15x+110)=7x(x-2)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \((-5x+5)(4x+4)-x(-24x-13)=11\\ \Leftrightarrow -20x^2-20x+20x+20 +24x^2+13x-11=0 \\ \Leftrightarrow 4x^2+13x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+13x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.4.9 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.4} & & = \frac{-13+\sqrt25}{2.4} \\ & = \frac{-18}{8} & & = \frac{-8}{8} \\ & = \frac{-9}{4} & & = -1 \\ \\ V &= \Big\{ \frac{-9}{4} ; -1 \Big\} & &\end{align} \\ -----------------\)
  2. \(3x^2-(20x-24)=2x(x-5) \\ \Leftrightarrow 3x^2-20x+24=2x^2-10x \\ \Leftrightarrow x^2-10x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.24 & &\\ & = 100-96 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-10)-\sqrt4}{2.1} & & = \frac{-(-10)+\sqrt4}{2.1} \\ & = \frac{8}{2} & & = \frac{12}{2} \\ & = 4 & & = 6 \\ \\ V &= \Big\{ 4 ; 6 \Big\} & &\end{align} \\ -----------------\)
  3. \(-\frac{1}{2}x=-\frac{1}{4}x^2-4 \\ \Leftrightarrow \frac{1}{4}x^2-\frac{1}{2}x+4=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{1}{2}x+4\right)=0 \color{red}{.4} \\ \Leftrightarrow 4x^2-8x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-8x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.4.64 & &\\ & = 64-1024 & & \\ & = -960 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  4. \(-(5-9x)=-48x^2-(2-2x) \\ \Leftrightarrow -5+9x=-48x^2-2+2x \\ \Leftrightarrow 48x^2+7x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
  5. \(-(2-44x)=-8x^2-(20-19x) \\ \Leftrightarrow -2+44x=-8x^2-20+19x \\ \Leftrightarrow 8x^2+25x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+25x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.8.18 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.8} & & = \frac{-25+\sqrt49}{2.8} \\ & = \frac{-32}{16} & & = \frac{-18}{16} \\ & = -2 & & = \frac{-9}{8} \\ \\ V &= \Big\{ -2 ; \frac{-9}{8} \Big\} & &\end{align} \\ -----------------\)
  6. \(\frac{16}{9}x^2+\frac{2}{3}x+9=0\\ \Leftrightarrow \color{red}{9.} \left(\frac{16}{9}x^2+\frac{2}{3}x+9\right)=0 \color{red}{.9} \\ \Leftrightarrow 16x^2+6x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+6x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.16.81 & &\\ & = 36-5184 & & \\ & = -5148 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  7. \(\frac{4}{3}x^2+\frac{5}{3}x-3=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{4}{3}x^2+\frac{5}{3}x-3\right)=0 \color{red}{.3} \\ \Leftrightarrow 4x^2+5x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.(-9) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.4} & & = \frac{-5+\sqrt169}{2.4} \\ & = \frac{-18}{8} & & = \frac{8}{8} \\ & = \frac{-9}{4} & & = 1 \\ \\ V &= \Big\{ \frac{-9}{4} ; 1 \Big\} & &\end{align} \\ -----------------\)
  8. \(x=-\frac{1}{3}x^2-\frac{4}{3} \\ \Leftrightarrow \frac{1}{3}x^2+x+\frac{4}{3}=0 \\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+x+\frac{4}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow 4x^2+12x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+12x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.4.16 & &\\ & = 144-256 & & \\ & = -112 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  9. \(12x^2+\frac{7}{6}x-\frac{1}{3}=0\\ \Leftrightarrow \color{red}{6.} \left(12x^2+\frac{7}{6}x-\frac{1}{3}\right)=0 \color{red}{.6} \\ \Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.72.(-2) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\ & = \frac{-32}{144} & & = \frac{18}{144} \\ & = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  10. \(-(14-8x)=-x^2-(-21-6x) \\ \Leftrightarrow -14+8x=-x^2+21+6x \\ \Leftrightarrow x^2+2x-35=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-35=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-35) & &\\ & = 4+140 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt144}{2.1} & & = \frac{-2+\sqrt144}{2.1} \\ & = \frac{-14}{2} & & = \frac{10}{2} \\ & = -7 & & = 5 \\ \\ V &= \Big\{ -7 ; 5 \Big\} & &\end{align} \\ -----------------\)
  11. \(x(x-120)=-100(x+1) \\ \Leftrightarrow x^2-120x=-100x-100 \\ \Leftrightarrow x^2-20x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-20x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-20)^2-4.1.100 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-20)}{2.1} & & \\ & = 10 & & \\V &= \Big\{ 10 \Big\} & &\end{align} \\ -----------------\)
  12. \(8x^2-(15x+110)=7x(x-2) \\ \Leftrightarrow 8x^2-15x-110=7x^2-14x \\ \Leftrightarrow x^2-x-110=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-110=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-110) & &\\ & = 1+440 & & \\ & = 441 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt441}{2.1} & & = \frac{-(-1)+\sqrt441}{2.1} \\ & = \frac{-20}{2} & & = \frac{22}{2} \\ & = -10 & & = 11 \\ \\ V &= \Big\{ -10 ; 11 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-03-14 05:35:48
Een site van Busleyden Atheneum Mechelen