VKV met breuken of rekenwerk

Hoofdmenu Eentje per keer 

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-(7-24x)=-18x^2-(-1-17x)\)
  2. \((-2x-5)(3x-1)-x(-30x+0)=11\)
  3. \(\frac{1}{2}x^2+\frac{5}{2}x-18=0\)
  4. \(\frac{3}{5}x=-\frac{8}{5}x^2+\frac{1}{10}\)
  5. \(\frac{1}{5}x^2+\frac{1}{2}x+\frac{1}{5}=0\)
  6. \(\frac{1}{4}x^2+\frac{15}{8}x-1=0\)
  7. \((5x+4)(-3x+5)-x(-16x+46)=130\)
  8. \(-(4-22x)=-3x^2-(-8-17x)\)
  9. \(3x=-\frac{1}{3}x^2-\frac{20}{3}\)
  10. \(4x^2-(3x-48)=x(x-28)\)
  11. \(-(12+5x)=-x^2-(82-12x)\)
  12. \(-\frac{3}{2}x=-\frac{1}{4}x^2-4\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-(7-24x)=-18x^2-(-1-17x) \\ \Leftrightarrow -7+24x=-18x^2+1+17x \\ \Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  2. \((-2x-5)(3x-1)-x(-30x+0)=11\\ \Leftrightarrow -6x^2+2x-15x+5 +30x^2+0x-11=0 \\ \Leftrightarrow 24x^2+7x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+7x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.24.(-6) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.24} & & = \frac{-7+\sqrt625}{2.24} \\ & = \frac{-32}{48} & & = \frac{18}{48} \\ & = \frac{-2}{3} & & = \frac{3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{3}{8} \Big\} & &\end{align} \\ -----------------\)
  3. \(\frac{1}{2}x^2+\frac{5}{2}x-18=0\\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+\frac{5}{2}x-18\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2+5x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-36) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.1} & & = \frac{-5+\sqrt169}{2.1} \\ & = \frac{-18}{2} & & = \frac{8}{2} \\ & = -9 & & = 4 \\ \\ V &= \Big\{ -9 ; 4 \Big\} & &\end{align} \\ -----------------\)
  4. \(\frac{3}{5}x=-\frac{8}{5}x^2+\frac{1}{10} \\ \Leftrightarrow \frac{8}{5}x^2+\frac{3}{5}x-\frac{1}{10}=0 \\ \Leftrightarrow \color{red}{10.} \left(\frac{8}{5}x^2+\frac{3}{5}x-\frac{1}{10}\right)=0 \color{red}{.10} \\ \Leftrightarrow 16x^2+6x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+6x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.16.(-1) & &\\ & = 36+64 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt100}{2.16} & & = \frac{-6+\sqrt100}{2.16} \\ & = \frac{-16}{32} & & = \frac{4}{32} \\ & = \frac{-1}{2} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{1}{5}x^2+\frac{1}{2}x+\frac{1}{5}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{1}{5}x^2+\frac{1}{2}x+\frac{1}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
  6. \(\frac{1}{4}x^2+\frac{15}{8}x-1=0\\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{4}x^2+\frac{15}{8}x-1\right)=0 \color{red}{.8} \\ \Leftrightarrow 2x^2+15x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+15x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.2.(-8) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.2} & & = \frac{-15+\sqrt289}{2.2} \\ & = \frac{-32}{4} & & = \frac{2}{4} \\ & = -8 & & = \frac{1}{2} \\ \\ V &= \Big\{ -8 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  7. \((5x+4)(-3x+5)-x(-16x+46)=130\\ \Leftrightarrow -15x^2+25x-12x+20 +16x^2-46x-130=0 \\ \Leftrightarrow x^2-x-110=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-110=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-110) & &\\ & = 1+440 & & \\ & = 441 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt441}{2.1} & & = \frac{-(-1)+\sqrt441}{2.1} \\ & = \frac{-20}{2} & & = \frac{22}{2} \\ & = -10 & & = 11 \\ \\ V &= \Big\{ -10 ; 11 \Big\} & &\end{align} \\ -----------------\)
  8. \(-(4-22x)=-3x^2-(-8-17x) \\ \Leftrightarrow -4+22x=-3x^2+8+17x \\ \Leftrightarrow 3x^2+5x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+5x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.3.(-12) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.3} & & = \frac{-5+\sqrt169}{2.3} \\ & = \frac{-18}{6} & & = \frac{8}{6} \\ & = -3 & & = \frac{4}{3} \\ \\ V &= \Big\{ -3 ; \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
  9. \(3x=-\frac{1}{3}x^2-\frac{20}{3} \\ \Leftrightarrow \frac{1}{3}x^2+3x+\frac{20}{3}=0 \\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+3x+\frac{20}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2+9x+20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+9x+20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (9)^2-4.1.20 & &\\ & = 81-80 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-9-\sqrt1}{2.1} & & = \frac{-9+\sqrt1}{2.1} \\ & = \frac{-10}{2} & & = \frac{-8}{2} \\ & = -5 & & = -4 \\ \\ V &= \Big\{ -5 ; -4 \Big\} & &\end{align} \\ -----------------\)
  10. \(4x^2-(3x-48)=x(x-28) \\ \Leftrightarrow 4x^2-3x+48=x^2-28x \\ \Leftrightarrow 3x^2+25x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+25x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.3.48 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.3} & & = \frac{-25+\sqrt49}{2.3} \\ & = \frac{-32}{6} & & = \frac{-18}{6} \\ & = \frac{-16}{3} & & = -3 \\ \\ V &= \Big\{ \frac{-16}{3} ; -3 \Big\} & &\end{align} \\ -----------------\)
  11. \(-(12+5x)=-x^2-(82-12x) \\ \Leftrightarrow -12-5x=-x^2-82+12x \\ \Leftrightarrow x^2-17x+70=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-17x+70=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-17)^2-4.1.70 & &\\ & = 289-280 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-17)-\sqrt9}{2.1} & & = \frac{-(-17)+\sqrt9}{2.1} \\ & = \frac{14}{2} & & = \frac{20}{2} \\ & = 7 & & = 10 \\ \\ V &= \Big\{ 7 ; 10 \Big\} & &\end{align} \\ -----------------\)
  12. \(-\frac{3}{2}x=-\frac{1}{4}x^2-4 \\ \Leftrightarrow \frac{1}{4}x^2-\frac{3}{2}x+4=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{3}{2}x+4\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-6x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.16 & &\\ & = 36-64 & & \\ & = -28 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-03-11 18:30:19
Een site van Busleyden Atheneum Mechelen