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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{1}{3}x^2+\frac{20}{3}x+33=0\)
  2. \(2x^2-(6x-121)=x(x+16)\)
  3. \(-(6-5x)=-x^2-(-1-11x)\)
  4. \(-\frac{5}{3}x=-\frac{2}{3}x^2-\frac{49}{24}\)
  5. \(\frac{1}{4}x^2+\frac{3}{4}x-10=0\)
  6. \(x(9x-25)=3(x-12)\)
  7. \(\frac{5}{2}x=-\frac{9}{2}x^2+2\)
  8. \((4x+2)(x-3)-x(-12x-33)=-5\)
  9. \(-\frac{1}{2}x=-\frac{1}{4}x^2+\frac{99}{4}\)
  10. \(\frac{9}{5}x^2+\frac{26}{5}x+\frac{49}{5}=0\)
  11. \((3x+2)(-3x+4)-x(-13x+15)=17\)
  12. \(x=-\frac{4}{3}x^2+\frac{1}{3}\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{1}{3}x^2+\frac{20}{3}x+33=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+\frac{20}{3}x+33\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2+20x+99=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+20x+99=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.1.99 & &\\ & = 400-396 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-20-\sqrt4}{2.1} & & = \frac{-20+\sqrt4}{2.1} \\ & = \frac{-22}{2} & & = \frac{-18}{2} \\ & = -11 & & = -9 \\ \\ V &= \Big\{ -11 ; -9 \Big\} & &\end{align} \\ -----------------\)
  2. \(2x^2-(6x-121)=x(x+16) \\ \Leftrightarrow 2x^2-6x+121=x^2+16x \\ \Leftrightarrow x^2-22x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-22x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.1.121 & &\\ & = 484-484 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-22)}{2.1} & & \\ & = 11 & & \\V &= \Big\{ 11 \Big\} & &\end{align} \\ -----------------\)
  3. \(-(6-5x)=-x^2-(-1-11x) \\ \Leftrightarrow -6+5x=-x^2+1+11x \\ \Leftrightarrow x^2-6x-7=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-7=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.(-7) & &\\ & = 36+28 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt64}{2.1} & & = \frac{-(-6)+\sqrt64}{2.1} \\ & = \frac{-2}{2} & & = \frac{14}{2} \\ & = -1 & & = 7 \\ \\ V &= \Big\{ -1 ; 7 \Big\} & &\end{align} \\ -----------------\)
  4. \(-\frac{5}{3}x=-\frac{2}{3}x^2-\frac{49}{24} \\ \Leftrightarrow \frac{2}{3}x^2-\frac{5}{3}x+\frac{49}{24}=0 \\ \Leftrightarrow \color{red}{24.} \left(\frac{2}{3}x^2-\frac{5}{3}x+\frac{49}{24}\right)=0 \color{red}{.24} \\ \Leftrightarrow 16x^2-40x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-40x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-40)^2-4.16.49 & &\\ & = 1600-3136 & & \\ & = -1536 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  5. \(\frac{1}{4}x^2+\frac{3}{4}x-10=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{3}{4}x-10\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+3x-40=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-40) & &\\ & = 9+160 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt169}{2.1} & & = \frac{-3+\sqrt169}{2.1} \\ & = \frac{-16}{2} & & = \frac{10}{2} \\ & = -8 & & = 5 \\ \\ V &= \Big\{ -8 ; 5 \Big\} & &\end{align} \\ -----------------\)
  6. \(x(9x-25)=3(x-12) \\ \Leftrightarrow 9x^2-25x=3x-36 \\ \Leftrightarrow 9x^2-28x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-28x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-28)^2-4.9.36 & &\\ & = 784-1296 & & \\ & = -512 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  7. \(\frac{5}{2}x=-\frac{9}{2}x^2+2 \\ \Leftrightarrow \frac{9}{2}x^2+\frac{5}{2}x-2=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{9}{2}x^2+\frac{5}{2}x-2\right)=0 \color{red}{.2} \\ \Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.9.(-4) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\ & = \frac{-18}{18} & & = \frac{8}{18} \\ & = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)
  8. \((4x+2)(x-3)-x(-12x-33)=-5\\ \Leftrightarrow 4x^2-12x+2x-6 +12x^2+33x+5=0 \\ \Leftrightarrow 16x^2+15x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+15x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.16.(-1) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.16} & & = \frac{-15+\sqrt289}{2.16} \\ & = \frac{-32}{32} & & = \frac{2}{32} \\ & = -1 & & = \frac{1}{16} \\ \\ V &= \Big\{ -1 ; \frac{1}{16} \Big\} & &\end{align} \\ -----------------\)
  9. \(-\frac{1}{2}x=-\frac{1}{4}x^2+\frac{99}{4} \\ \Leftrightarrow \frac{1}{4}x^2-\frac{1}{2}x-\frac{99}{4}=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{1}{2}x-\frac{99}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-2x-99=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-99=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-99) & &\\ & = 4+396 & & \\ & = 400 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt400}{2.1} & & = \frac{-(-2)+\sqrt400}{2.1} \\ & = \frac{-18}{2} & & = \frac{22}{2} \\ & = -9 & & = 11 \\ \\ V &= \Big\{ -9 ; 11 \Big\} & &\end{align} \\ -----------------\)
  10. \(\frac{9}{5}x^2+\frac{26}{5}x+\frac{49}{5}=0\\ \Leftrightarrow \color{red}{5.} \left(\frac{9}{5}x^2+\frac{26}{5}x+\frac{49}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow 9x^2+26x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+26x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (26)^2-4.9.49 & &\\ & = 676-1764 & & \\ & = -1088 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  11. \((3x+2)(-3x+4)-x(-13x+15)=17\\ \Leftrightarrow -9x^2+12x-6x+8 +13x^2-15x-17=0 \\ \Leftrightarrow 4x^2+5x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.(-9) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.4} & & = \frac{-5+\sqrt169}{2.4} \\ & = \frac{-18}{8} & & = \frac{8}{8} \\ & = \frac{-9}{4} & & = 1 \\ \\ V &= \Big\{ \frac{-9}{4} ; 1 \Big\} & &\end{align} \\ -----------------\)
  12. \(x=-\frac{4}{3}x^2+\frac{1}{3} \\ \Leftrightarrow \frac{4}{3}x^2+x-\frac{1}{3}=0 \\ \Leftrightarrow \color{red}{3.} \left(\frac{4}{3}x^2+x-\frac{1}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow 4x^2+3x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+3x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.4.(-1) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.4} & & = \frac{-3+\sqrt25}{2.4} \\ & = \frac{-8}{8} & & = \frac{2}{8} \\ & = -1 & & = \frac{1}{4} \\ \\ V &= \Big\{ -1 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2024-11-21 07:37:13
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