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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(17x^2-(13x-81)=x(x+59)\)
  2. \(\frac{7}{4}x=-\frac{9}{2}x^2+2\)
  3. \((-4x-4)(5x+4)-x(-21x-21)=-34\)
  4. \((-x-5)(-2x+2)-x(x-2)=1\)
  5. \(x=-\frac{1}{2}x^2+\frac{15}{2}\)
  6. \(\frac{4}{3}x=-\frac{1}{3}x^2+15\)
  7. \(-(3-x)=-x^2-(73-18x)\)
  8. \(10x^2-(8x-4)=x(x-14)\)
  9. \(\frac{5}{2}x=-2x^2+\frac{9}{2}\)
  10. \(15x^2-(4x-2)=7x(x-3)\)
  11. \(x(18x+7)=2(x+1)\)
  12. \(x(9x+63)=9(x-9)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(17x^2-(13x-81)=x(x+59) \\ \Leftrightarrow 17x^2-13x+81=x^2+59x \\ \Leftrightarrow 16x^2-72x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-72x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-72)^2-4.16.81 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-72)}{2.16} & & \\ & = \frac{9}{4} & & \\V &= \Big\{ \frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
  2. \(\frac{7}{4}x=-\frac{9}{2}x^2+2 \\ \Leftrightarrow \frac{9}{2}x^2+\frac{7}{4}x-2=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{9}{2}x^2+\frac{7}{4}x-2\right)=0 \color{red}{.4} \\ \Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  3. \((-4x-4)(5x+4)-x(-21x-21)=-34\\ \Leftrightarrow -20x^2-16x-20x-16 +21x^2+21x+34=0 \\ \Leftrightarrow x^2-11x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-11)^2-4.1.18 & &\\ & = 121-72 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-11)-\sqrt49}{2.1} & & = \frac{-(-11)+\sqrt49}{2.1} \\ & = \frac{4}{2} & & = \frac{18}{2} \\ & = 2 & & = 9 \\ \\ V &= \Big\{ 2 ; 9 \Big\} & &\end{align} \\ -----------------\)
  4. \((-x-5)(-2x+2)-x(x-2)=1\\ \Leftrightarrow 2x^2-2x+10x-10 -x^2+2x-1=0 \\ \Leftrightarrow x^2-10x-11=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x-11=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.(-11) & &\\ & = 100+44 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-10)-\sqrt144}{2.1} & & = \frac{-(-10)+\sqrt144}{2.1} \\ & = \frac{-2}{2} & & = \frac{22}{2} \\ & = -1 & & = 11 \\ \\ V &= \Big\{ -1 ; 11 \Big\} & &\end{align} \\ -----------------\)
  5. \(x=-\frac{1}{2}x^2+\frac{15}{2} \\ \Leftrightarrow \frac{1}{2}x^2+x-\frac{15}{2}=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+x-\frac{15}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2+2x-15=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-15=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-15) & &\\ & = 4+60 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt64}{2.1} & & = \frac{-2+\sqrt64}{2.1} \\ & = \frac{-10}{2} & & = \frac{6}{2} \\ & = -5 & & = 3 \\ \\ V &= \Big\{ -5 ; 3 \Big\} & &\end{align} \\ -----------------\)
  6. \(\frac{4}{3}x=-\frac{1}{3}x^2+15 \\ \Leftrightarrow \frac{1}{3}x^2+\frac{4}{3}x-15=0 \\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+\frac{4}{3}x-15\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2+4x-45=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-45=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-45) & &\\ & = 16+180 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt196}{2.1} & & = \frac{-4+\sqrt196}{2.1} \\ & = \frac{-18}{2} & & = \frac{10}{2} \\ & = -9 & & = 5 \\ \\ V &= \Big\{ -9 ; 5 \Big\} & &\end{align} \\ -----------------\)
  7. \(-(3-x)=-x^2-(73-18x) \\ \Leftrightarrow -3+x=-x^2-73+18x \\ \Leftrightarrow x^2-17x+70=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-17x+70=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-17)^2-4.1.70 & &\\ & = 289-280 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-17)-\sqrt9}{2.1} & & = \frac{-(-17)+\sqrt9}{2.1} \\ & = \frac{14}{2} & & = \frac{20}{2} \\ & = 7 & & = 10 \\ \\ V &= \Big\{ 7 ; 10 \Big\} & &\end{align} \\ -----------------\)
  8. \(10x^2-(8x-4)=x(x-14) \\ \Leftrightarrow 10x^2-8x+4=x^2-14x \\ \Leftrightarrow 9x^2+6x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+6x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.9.4 & &\\ & = 36-144 & & \\ & = -108 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  9. \(\frac{5}{2}x=-2x^2+\frac{9}{2} \\ \Leftrightarrow 2x^2+\frac{5}{2}x-\frac{9}{2}=0 \\ \Leftrightarrow \color{red}{2.} \left(2x^2+\frac{5}{2}x-\frac{9}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow 4x^2+5x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.(-9) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.4} & & = \frac{-5+\sqrt169}{2.4} \\ & = \frac{-18}{8} & & = \frac{8}{8} \\ & = \frac{-9}{4} & & = 1 \\ \\ V &= \Big\{ \frac{-9}{4} ; 1 \Big\} & &\end{align} \\ -----------------\)
  10. \(15x^2-(4x-2)=7x(x-3) \\ \Leftrightarrow 15x^2-4x+2=7x^2-21x \\ \Leftrightarrow 8x^2+17x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+17x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.8.2 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.8} & & = \frac{-17+\sqrt225}{2.8} \\ & = \frac{-32}{16} & & = \frac{-2}{16} \\ & = -2 & & = \frac{-1}{8} \\ \\ V &= \Big\{ -2 ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
  11. \(x(18x+7)=2(x+1) \\ \Leftrightarrow 18x^2+7x=2x+2 \\ \Leftrightarrow 18x^2+5x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+5x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.18.(-2) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.18} & & = \frac{-5+\sqrt169}{2.18} \\ & = \frac{-18}{36} & & = \frac{8}{36} \\ & = \frac{-1}{2} & & = \frac{2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{2}{9} \Big\} & &\end{align} \\ -----------------\)
  12. \(x(9x+63)=9(x-9) \\ \Leftrightarrow 9x^2+63x=9x-81 \\ \Leftrightarrow 9x^2+54x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+54x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (54)^2-4.9.81 & &\\ & = 2916-2916 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-54}{2.9} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator vanhoeckes.be/wiskunde 2025-05-19 06:13:59