VKV met breuken of rekenwerk

Hoofdmenu Eentje per keer 

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(2x^2-(12x+20)=x(x-4)\)
  2. \(x(16x-53)=5(x-20)\)
  3. \(7x^2-(10x-1)=3x(x-5)\)
  4. \(24x^2-(17x-2)=6x(x-5)\)
  5. \(x(4x-21)=7(x-7)\)
  6. \(x(x+13)=2(x-14)\)
  7. \(\frac{1}{2}x^2+\frac{5}{12}x-\frac{1}{2}=0\)
  8. \(6x^2-(19x-8)=5x(x-2)\)
  9. \(-(4-15x)=-x^2-(-38-14x)\)
  10. \(x(x-9)=6(x-6)\)
  11. \(3x^2-(6x-4)=2x(x-5)\)
  12. \((2x+4)(-3x-5)-x(-22x-6)=-29\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(2x^2-(12x+20)=x(x-4) \\ \Leftrightarrow 2x^2-12x-20=x^2-4x \\ \Leftrightarrow x^2-8x-20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-20) & &\\ & = 64+80 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt144}{2.1} & & = \frac{-(-8)+\sqrt144}{2.1} \\ & = \frac{-4}{2} & & = \frac{20}{2} \\ & = -2 & & = 10 \\ \\ V &= \Big\{ -2 ; 10 \Big\} & &\end{align} \\ -----------------\)
  2. \(x(16x-53)=5(x-20) \\ \Leftrightarrow 16x^2-53x=5x-100 \\ \Leftrightarrow 16x^2-58x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-58x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-58)^2-4.16.100 & &\\ & = 3364-6400 & & \\ & = -3036 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  3. \(7x^2-(10x-1)=3x(x-5) \\ \Leftrightarrow 7x^2-10x+1=3x^2-15x \\ \Leftrightarrow 4x^2+5x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.1 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.4} & & = \frac{-5+\sqrt9}{2.4} \\ & = \frac{-8}{8} & & = \frac{-2}{8} \\ & = -1 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -1 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  4. \(24x^2-(17x-2)=6x(x-5) \\ \Leftrightarrow 24x^2-17x+2=6x^2-30x \\ \Leftrightarrow 18x^2+13x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+13x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.18.2 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.18} & & = \frac{-13+\sqrt25}{2.18} \\ & = \frac{-18}{36} & & = \frac{-8}{36} \\ & = \frac{-1}{2} & & = \frac{-2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-2}{9} \Big\} & &\end{align} \\ -----------------\)
  5. \(x(4x-21)=7(x-7) \\ \Leftrightarrow 4x^2-21x=7x-49 \\ \Leftrightarrow 4x^2-28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-28x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-28)^2-4.4.49 & &\\ & = 784-784 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-28)}{2.4} & & \\ & = \frac{7}{2} & & \\V &= \Big\{ \frac{7}{2} \Big\} & &\end{align} \\ -----------------\)
  6. \(x(x+13)=2(x-14) \\ \Leftrightarrow x^2+13x=2x-28 \\ \Leftrightarrow x^2+11x+28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+28=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.28 & &\\ & = 121-112 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt9}{2.1} & & = \frac{-11+\sqrt9}{2.1} \\ & = \frac{-14}{2} & & = \frac{-8}{2} \\ & = -7 & & = -4 \\ \\ V &= \Big\{ -7 ; -4 \Big\} & &\end{align} \\ -----------------\)
  7. \(\frac{1}{2}x^2+\frac{5}{12}x-\frac{1}{2}=0\\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{2}x^2+\frac{5}{12}x-\frac{1}{2}\right)=0 \color{red}{.12} \\ \Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
  8. \(6x^2-(19x-8)=5x(x-2) \\ \Leftrightarrow 6x^2-19x+8=5x^2-10x \\ \Leftrightarrow x^2-9x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-9)^2-4.1.8 & &\\ & = 81-32 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-9)-\sqrt49}{2.1} & & = \frac{-(-9)+\sqrt49}{2.1} \\ & = \frac{2}{2} & & = \frac{16}{2} \\ & = 1 & & = 8 \\ \\ V &= \Big\{ 1 ; 8 \Big\} & &\end{align} \\ -----------------\)
  9. \(-(4-15x)=-x^2-(-38-14x) \\ \Leftrightarrow -4+15x=-x^2+38+14x \\ \Leftrightarrow x^2+x-42=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-42=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-42) & &\\ & = 1+168 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt169}{2.1} & & = \frac{-1+\sqrt169}{2.1} \\ & = \frac{-14}{2} & & = \frac{12}{2} \\ & = -7 & & = 6 \\ \\ V &= \Big\{ -7 ; 6 \Big\} & &\end{align} \\ -----------------\)
  10. \(x(x-9)=6(x-6) \\ \Leftrightarrow x^2-9x=6x-36 \\ \Leftrightarrow x^2-15x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-15x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-15)^2-4.1.36 & &\\ & = 225-144 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-15)-\sqrt81}{2.1} & & = \frac{-(-15)+\sqrt81}{2.1} \\ & = \frac{6}{2} & & = \frac{24}{2} \\ & = 3 & & = 12 \\ \\ V &= \Big\{ 3 ; 12 \Big\} & &\end{align} \\ -----------------\)
  11. \(3x^2-(6x-4)=2x(x-5) \\ \Leftrightarrow 3x^2-6x+4=2x^2-10x \\ \Leftrightarrow x^2+4x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.1} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
  12. \((2x+4)(-3x-5)-x(-22x-6)=-29\\ \Leftrightarrow -6x^2-10x-12x-20 +22x^2+6x+29=0 \\ \Leftrightarrow 16x^2-24x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-24x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.16.9 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-24)}{2.16} & & \\ & = \frac{3}{4} & & \\V &= \Big\{ \frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-02-18 23:28:18
Een site van Busleyden Atheneum Mechelen