VKV met breuken of rekenwerk

Hoofdmenu Eentje per keer 

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(20x^2-(9x-48)=17x(x-2)\)
  2. \(3x^2-(20x-33)=2x(x-17)\)
  3. \((-2x+3)(-4x+3)-x(7x+12)=-11\)
  4. \(x(x+55)=50(x+1)\)
  5. \(\frac{1}{5}x=-\frac{1}{100}x^2-1\)
  6. \((-4x-2)(-3x+4)-x(-4x-120)=-152\)
  7. \(\frac{7}{5}x=-\frac{1}{5}x^2+6\)
  8. \(-(3-9x)=-x^2-(-11-14x)\)
  9. \(7x^2-(17x+48)=4x(x-6)\)
  10. \(2x^2-(18x+96)=x(x-14)\)
  11. \((-5x-1)(2x+3)-x(-19x-24)=-39\)
  12. \(\frac{1}{11}x^2-x+\frac{11}{4}=0\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(20x^2-(9x-48)=17x(x-2) \\ \Leftrightarrow 20x^2-9x+48=17x^2-34x \\ \Leftrightarrow 3x^2+25x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+25x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.3.48 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.3} & & = \frac{-25+\sqrt49}{2.3} \\ & = \frac{-32}{6} & & = \frac{-18}{6} \\ & = \frac{-16}{3} & & = -3 \\ \\ V &= \Big\{ \frac{-16}{3} ; -3 \Big\} & &\end{align} \\ -----------------\)
  2. \(3x^2-(20x-33)=2x(x-17) \\ \Leftrightarrow 3x^2-20x+33=2x^2-34x \\ \Leftrightarrow x^2+14x+33=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+33=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.33 & &\\ & = 196-132 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-14-\sqrt64}{2.1} & & = \frac{-14+\sqrt64}{2.1} \\ & = \frac{-22}{2} & & = \frac{-6}{2} \\ & = -11 & & = -3 \\ \\ V &= \Big\{ -11 ; -3 \Big\} & &\end{align} \\ -----------------\)
  3. \((-2x+3)(-4x+3)-x(7x+12)=-11\\ \Leftrightarrow 8x^2-6x-12x+9 -7x^2-12x+11=0 \\ \Leftrightarrow x^2-9x+20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x+20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-9)^2-4.1.20 & &\\ & = 81-80 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-9)-\sqrt1}{2.1} & & = \frac{-(-9)+\sqrt1}{2.1} \\ & = \frac{8}{2} & & = \frac{10}{2} \\ & = 4 & & = 5 \\ \\ V &= \Big\{ 4 ; 5 \Big\} & &\end{align} \\ -----------------\)
  4. \(x(x+55)=50(x+1) \\ \Leftrightarrow x^2+55x=50x+50 \\ \Leftrightarrow x^2+5x-50=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-50=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-50) & &\\ & = 25+200 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt225}{2.1} & & = \frac{-5+\sqrt225}{2.1} \\ & = \frac{-20}{2} & & = \frac{10}{2} \\ & = -10 & & = 5 \\ \\ V &= \Big\{ -10 ; 5 \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{1}{5}x=-\frac{1}{100}x^2-1 \\ \Leftrightarrow \frac{1}{100}x^2+\frac{1}{5}x+1=0 \\ \Leftrightarrow \color{red}{100.} \left(\frac{1}{100}x^2+\frac{1}{5}x+1\right)=0 \color{red}{.100} \\ \Leftrightarrow x^2+20x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+20x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.1.100 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.1} & & \\ & = -10 & & \\V &= \Big\{ -10 \Big\} & &\end{align} \\ -----------------\)
  6. \((-4x-2)(-3x+4)-x(-4x-120)=-152\\ \Leftrightarrow 12x^2-16x+6x-8 +4x^2+120x+152=0 \\ \Leftrightarrow 16x^2+96x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+96x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (96)^2-4.16.144 & &\\ & = 9216-9216 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-96}{2.16} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
  7. \(\frac{7}{5}x=-\frac{1}{5}x^2+6 \\ \Leftrightarrow \frac{1}{5}x^2+\frac{7}{5}x-6=0 \\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+\frac{7}{5}x-6\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2+7x-30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+7x-30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.(-30) & &\\ & = 49+120 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt169}{2.1} & & = \frac{-7+\sqrt169}{2.1} \\ & = \frac{-20}{2} & & = \frac{6}{2} \\ & = -10 & & = 3 \\ \\ V &= \Big\{ -10 ; 3 \Big\} & &\end{align} \\ -----------------\)
  8. \(-(3-9x)=-x^2-(-11-14x) \\ \Leftrightarrow -3+9x=-x^2+11+14x \\ \Leftrightarrow x^2-5x-14=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-14=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-14) & &\\ & = 25+56 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt81}{2.1} & & = \frac{-(-5)+\sqrt81}{2.1} \\ & = \frac{-4}{2} & & = \frac{14}{2} \\ & = -2 & & = 7 \\ \\ V &= \Big\{ -2 ; 7 \Big\} & &\end{align} \\ -----------------\)
  9. \(7x^2-(17x+48)=4x(x-6) \\ \Leftrightarrow 7x^2-17x-48=4x^2-24x \\ \Leftrightarrow 3x^2+7x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+7x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.3.(-48) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.3} & & = \frac{-7+\sqrt625}{2.3} \\ & = \frac{-32}{6} & & = \frac{18}{6} \\ & = \frac{-16}{3} & & = 3 \\ \\ V &= \Big\{ \frac{-16}{3} ; 3 \Big\} & &\end{align} \\ -----------------\)
  10. \(2x^2-(18x+96)=x(x-14) \\ \Leftrightarrow 2x^2-18x-96=x^2-14x \\ \Leftrightarrow x^2-4x-96=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-96=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-96) & &\\ & = 16+384 & & \\ & = 400 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt400}{2.1} & & = \frac{-(-4)+\sqrt400}{2.1} \\ & = \frac{-16}{2} & & = \frac{24}{2} \\ & = -8 & & = 12 \\ \\ V &= \Big\{ -8 ; 12 \Big\} & &\end{align} \\ -----------------\)
  11. \((-5x-1)(2x+3)-x(-19x-24)=-39\\ \Leftrightarrow -10x^2-15x-2x-3 +19x^2+24x+39=0 \\ \Leftrightarrow 9x^2+6x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+6x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.9.36 & &\\ & = 36-1296 & & \\ & = -1260 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  12. \(\frac{1}{11}x^2-x+\frac{11}{4}=0\\ \Leftrightarrow \color{red}{44.} \left(\frac{1}{11}x^2-x+\frac{11}{4}\right)=0 \color{red}{.44} \\ \Leftrightarrow 4x^2-44x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-44x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-44)^2-4.4.121 & &\\ & = 1936-1936 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-44)}{2.4} & & \\ & = \frac{11}{2} & & \\V &= \Big\{ \frac{11}{2} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-02-18 11:21:34
Een site van Busleyden Atheneum Mechelen