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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{1}{9}x^2+\frac{2}{3}x+1=0\)
  2. \((3x-5)(5x+5)-x(14x-5)=41\)
  3. \(-4x=-\frac{1}{2}x^2+\frac{33}{2}\)
  4. \((2x-2)(-2x-2)-x(-13x+6)=3\)
  5. \(x(4x-77)=-49(x+1)\)
  6. \((4x+4)(-3x+4)-x(-13x+27)=100\)
  7. \(-\frac{2}{3}x=-\frac{1}{12}x^2+\frac{5}{3}\)
  8. \((x+2)(2x-1)-x(-14x-75)=-83\)
  9. \(\frac{1}{4}x^2-\frac{1}{2}x+\frac{1}{4}=0\)
  10. \((5x-1)(-4x+2)-x(-24x-17)=-38\)
  11. \(\frac{1}{4}x^2+\frac{7}{2}x+\frac{45}{4}=0\)
  12. \(\frac{9}{2}x^2+\frac{13}{2}x+2=0\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{1}{9}x^2+\frac{2}{3}x+1=0\\ \Leftrightarrow \color{red}{9.} \left(\frac{1}{9}x^2+\frac{2}{3}x+1\right)=0 \color{red}{.9} \\ \Leftrightarrow 4x^2+24x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+24x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.4.36 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.4} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
  2. \((3x-5)(5x+5)-x(14x-5)=41\\ \Leftrightarrow 15x^2+15x-25x-25 -14x^2+5x-41=0 \\ \Leftrightarrow x^2-5x-66=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-66=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-66) & &\\ & = 25+264 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt289}{2.1} & & = \frac{-(-5)+\sqrt289}{2.1} \\ & = \frac{-12}{2} & & = \frac{22}{2} \\ & = -6 & & = 11 \\ \\ V &= \Big\{ -6 ; 11 \Big\} & &\end{align} \\ -----------------\)
  3. \(-4x=-\frac{1}{2}x^2+\frac{33}{2} \\ \Leftrightarrow \frac{1}{2}x^2-4x-\frac{33}{2}=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-4x-\frac{33}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2-8x-33=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-33=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-33) & &\\ & = 64+132 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt196}{2.1} & & = \frac{-(-8)+\sqrt196}{2.1} \\ & = \frac{-6}{2} & & = \frac{22}{2} \\ & = -3 & & = 11 \\ \\ V &= \Big\{ -3 ; 11 \Big\} & &\end{align} \\ -----------------\)
  4. \((2x-2)(-2x-2)-x(-13x+6)=3\\ \Leftrightarrow -4x^2-4x+4x+4 +13x^2-6x-3=0 \\ \Leftrightarrow 9x^2-6x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-6x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.9.1 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.9} & & \\ & = \frac{1}{3} & & \\V &= \Big\{ \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
  5. \(x(4x-77)=-49(x+1) \\ \Leftrightarrow 4x^2-77x=-49x-49 \\ \Leftrightarrow 4x^2-28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-28x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-28)^2-4.4.49 & &\\ & = 784-784 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-28)}{2.4} & & \\ & = \frac{7}{2} & & \\V &= \Big\{ \frac{7}{2} \Big\} & &\end{align} \\ -----------------\)
  6. \((4x+4)(-3x+4)-x(-13x+27)=100\\ \Leftrightarrow -12x^2+16x-12x+16 +13x^2-27x-100=0 \\ \Leftrightarrow x^2+5x-84=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-84=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-84) & &\\ & = 25+336 & & \\ & = 361 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt361}{2.1} & & = \frac{-5+\sqrt361}{2.1} \\ & = \frac{-24}{2} & & = \frac{14}{2} \\ & = -12 & & = 7 \\ \\ V &= \Big\{ -12 ; 7 \Big\} & &\end{align} \\ -----------------\)
  7. \(-\frac{2}{3}x=-\frac{1}{12}x^2+\frac{5}{3} \\ \Leftrightarrow \frac{1}{12}x^2-\frac{2}{3}x-\frac{5}{3}=0 \\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{12}x^2-\frac{2}{3}x-\frac{5}{3}\right)=0 \color{red}{.12} \\ \Leftrightarrow x^2-8x-20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-20) & &\\ & = 64+80 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt144}{2.1} & & = \frac{-(-8)+\sqrt144}{2.1} \\ & = \frac{-4}{2} & & = \frac{20}{2} \\ & = -2 & & = 10 \\ \\ V &= \Big\{ -2 ; 10 \Big\} & &\end{align} \\ -----------------\)
  8. \((x+2)(2x-1)-x(-14x-75)=-83\\ \Leftrightarrow 2x^2-x+4x-2 +14x^2+75x+83=0 \\ \Leftrightarrow 16x^2+72x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+72x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (72)^2-4.16.81 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-72}{2.16} & & \\ & = -\frac{9}{4} & & \\V &= \Big\{ -\frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
  9. \(\frac{1}{4}x^2-\frac{1}{2}x+\frac{1}{4}=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{1}{2}x+\frac{1}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-2x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.1 & &\\ & = 4-4 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-2)}{2.1} & & \\ & = 1 & & \\V &= \Big\{ 1 \Big\} & &\end{align} \\ -----------------\)
  10. \((5x-1)(-4x+2)-x(-24x-17)=-38\\ \Leftrightarrow -20x^2+10x+4x-2 +24x^2+17x+38=0 \\ \Leftrightarrow 4x^2+25x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+25x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.4.36 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.4} & & = \frac{-25+\sqrt49}{2.4} \\ & = \frac{-32}{8} & & = \frac{-18}{8} \\ & = -4 & & = \frac{-9}{4} \\ \\ V &= \Big\{ -4 ; \frac{-9}{4} \Big\} & &\end{align} \\ -----------------\)
  11. \(\frac{1}{4}x^2+\frac{7}{2}x+\frac{45}{4}=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{7}{2}x+\frac{45}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+14x+45=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+45=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.45 & &\\ & = 196-180 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-14-\sqrt16}{2.1} & & = \frac{-14+\sqrt16}{2.1} \\ & = \frac{-18}{2} & & = \frac{-10}{2} \\ & = -9 & & = -5 \\ \\ V &= \Big\{ -9 ; -5 \Big\} & &\end{align} \\ -----------------\)
  12. \(\frac{9}{2}x^2+\frac{13}{2}x+2=0\\ \Leftrightarrow \color{red}{2.} \left(\frac{9}{2}x^2+\frac{13}{2}x+2\right)=0 \color{red}{.2} \\ \Leftrightarrow 9x^2+13x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+13x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.9.4 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.9} & & = \frac{-13+\sqrt25}{2.9} \\ & = \frac{-18}{18} & & = \frac{-8}{18} \\ & = -1 & & = \frac{-4}{9} \\ \\ V &= \Big\{ -1 ; \frac{-4}{9} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator vanhoeckes.be/wiskunde 2025-04-03 05:16:21