VKV met breuken of rekenwerk

Hoofdmenu Eentje per keer 

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(2x^2-(11x-84)=x(x+8)\)
  2. \(-(9-23x)=-x^2-(-1-14x)\)
  3. \(x(x+27)=10(x-6)\)
  4. \(x(9x-14)=4(x-25)\)
  5. \(-(14-16x)=-x^2-(2-15x)\)
  6. \((x-4)(-2x-3)-x(-11x+31)=-13\)
  7. \(x(48x+22)=-3(x+1)\)
  8. \((4x-4)(2x-2)-x(-16x-7)=14\)
  9. \(\frac{9}{2}x=-\frac{1}{2}x^2-7\)
  10. \((4x+4)(-2x+3)-x(-26x+19)=14\)
  11. \(2x^2-(13x+24)=x(x-11)\)
  12. \(12x^2-(8x-40)=11x(x-2)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(2x^2-(11x-84)=x(x+8) \\ \Leftrightarrow 2x^2-11x+84=x^2+8x \\ \Leftrightarrow x^2-19x+84=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-19x+84=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-19)^2-4.1.84 & &\\ & = 361-336 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-19)-\sqrt25}{2.1} & & = \frac{-(-19)+\sqrt25}{2.1} \\ & = \frac{14}{2} & & = \frac{24}{2} \\ & = 7 & & = 12 \\ \\ V &= \Big\{ 7 ; 12 \Big\} & &\end{align} \\ -----------------\)
  2. \(-(9-23x)=-x^2-(-1-14x) \\ \Leftrightarrow -9+23x=-x^2+1+14x \\ \Leftrightarrow x^2+9x-10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+9x-10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (9)^2-4.1.(-10) & &\\ & = 81+40 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-9-\sqrt121}{2.1} & & = \frac{-9+\sqrt121}{2.1} \\ & = \frac{-20}{2} & & = \frac{2}{2} \\ & = -10 & & = 1 \\ \\ V &= \Big\{ -10 ; 1 \Big\} & &\end{align} \\ -----------------\)
  3. \(x(x+27)=10(x-6) \\ \Leftrightarrow x^2+27x=10x-60 \\ \Leftrightarrow x^2+17x+60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+17x+60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.1.60 & &\\ & = 289-240 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt49}{2.1} & & = \frac{-17+\sqrt49}{2.1} \\ & = \frac{-24}{2} & & = \frac{-10}{2} \\ & = -12 & & = -5 \\ \\ V &= \Big\{ -12 ; -5 \Big\} & &\end{align} \\ -----------------\)
  4. \(x(9x-14)=4(x-25) \\ \Leftrightarrow 9x^2-14x=4x-100 \\ \Leftrightarrow 9x^2-18x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-18x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-18)^2-4.9.100 & &\\ & = 324-3600 & & \\ & = -3276 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  5. \(-(14-16x)=-x^2-(2-15x) \\ \Leftrightarrow -14+16x=-x^2-2+15x \\ \Leftrightarrow x^2+x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-12) & &\\ & = 1+48 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt49}{2.1} & & = \frac{-1+\sqrt49}{2.1} \\ & = \frac{-8}{2} & & = \frac{6}{2} \\ & = -4 & & = 3 \\ \\ V &= \Big\{ -4 ; 3 \Big\} & &\end{align} \\ -----------------\)
  6. \((x-4)(-2x-3)-x(-11x+31)=-13\\ \Leftrightarrow -2x^2-3x+8x+12 +11x^2-31x+13=0 \\ \Leftrightarrow 9x^2-22x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-22x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.9.25 & &\\ & = 484-900 & & \\ & = -416 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  7. \(x(48x+22)=-3(x+1) \\ \Leftrightarrow 48x^2+22x=-3x-3 \\ \Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.48.3 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\ & = \frac{-32}{96} & & = \frac{-18}{96} \\ & = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
  8. \((4x-4)(2x-2)-x(-16x-7)=14\\ \Leftrightarrow 8x^2-8x-8x+8 +16x^2+7x-14=0 \\ \Leftrightarrow 24x^2+7x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+7x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.24.(-6) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.24} & & = \frac{-7+\sqrt625}{2.24} \\ & = \frac{-32}{48} & & = \frac{18}{48} \\ & = \frac{-2}{3} & & = \frac{3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{3}{8} \Big\} & &\end{align} \\ -----------------\)
  9. \(\frac{9}{2}x=-\frac{1}{2}x^2-7 \\ \Leftrightarrow \frac{1}{2}x^2+\frac{9}{2}x+7=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+\frac{9}{2}x+7\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2+9x+14=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+9x+14=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (9)^2-4.1.14 & &\\ & = 81-56 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-9-\sqrt25}{2.1} & & = \frac{-9+\sqrt25}{2.1} \\ & = \frac{-14}{2} & & = \frac{-4}{2} \\ & = -7 & & = -2 \\ \\ V &= \Big\{ -7 ; -2 \Big\} & &\end{align} \\ -----------------\)
  10. \((4x+4)(-2x+3)-x(-26x+19)=14\\ \Leftrightarrow -8x^2+12x-8x+12 +26x^2-19x-14=0 \\ \Leftrightarrow 18x^2+5x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+5x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.18.(-2) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.18} & & = \frac{-5+\sqrt169}{2.18} \\ & = \frac{-18}{36} & & = \frac{8}{36} \\ & = \frac{-1}{2} & & = \frac{2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{2}{9} \Big\} & &\end{align} \\ -----------------\)
  11. \(2x^2-(13x+24)=x(x-11) \\ \Leftrightarrow 2x^2-13x-24=x^2-11x \\ \Leftrightarrow x^2-2x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-24) & &\\ & = 4+96 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt100}{2.1} & & = \frac{-(-2)+\sqrt100}{2.1} \\ & = \frac{-8}{2} & & = \frac{12}{2} \\ & = -4 & & = 6 \\ \\ V &= \Big\{ -4 ; 6 \Big\} & &\end{align} \\ -----------------\)
  12. \(12x^2-(8x-40)=11x(x-2) \\ \Leftrightarrow 12x^2-8x+40=11x^2-22x \\ \Leftrightarrow x^2+14x+40=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.40 & &\\ & = 196-160 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-14-\sqrt36}{2.1} & & = \frac{-14+\sqrt36}{2.1} \\ & = \frac{-20}{2} & & = \frac{-8}{2} \\ & = -10 & & = -4 \\ \\ V &= \Big\{ -10 ; -4 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2025-11-01 21:01:17
Een site van Busleyden Atheneum Mechelen