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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{3}{4}x=-x^2-\frac{25}{4}\)
  2. \(-\frac{1}{2}x=-\frac{1}{8}x^2+4\)
  3. \((x+2)(5x-5)-x(4x-23)=-22\)
  4. \(10x^2-(11x-121)=x(x+55)\)
  5. \(\frac{1}{3}x^2+\frac{2}{3}x-21=0\)
  6. \(-(11-20x)=-72x^2-(9-13x)\)
  7. \(-(9-22x)=-x^2-(45-10x)\)
  8. \(-(3-17x)=-x^2-(52-3x)\)
  9. \(2x^2-(10x-42)=x(x+3)\)
  10. \(\frac{5}{24}x=-\frac{1}{4}x^2+\frac{1}{4}\)
  11. \((4x+4)(5x+4)-x(4x-48)=-84\)
  12. \(\frac{17}{4}x=-\frac{1}{2}x^2-2\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{3}{4}x=-x^2-\frac{25}{4} \\ \Leftrightarrow x^2+\frac{3}{4}x+\frac{25}{4}=0 \\ \Leftrightarrow \color{red}{4.} \left(x^2+\frac{3}{4}x+\frac{25}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow 16x^2+12x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+12x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.16.100 & &\\ & = 144-6400 & & \\ & = -6256 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  2. \(-\frac{1}{2}x=-\frac{1}{8}x^2+4 \\ \Leftrightarrow \frac{1}{8}x^2-\frac{1}{2}x-4=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2-\frac{1}{2}x-4\right)=0 \color{red}{.8} \\ \Leftrightarrow x^2-4x-32=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-32=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-32) & &\\ & = 16+128 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt144}{2.1} & & = \frac{-(-4)+\sqrt144}{2.1} \\ & = \frac{-8}{2} & & = \frac{16}{2} \\ & = -4 & & = 8 \\ \\ V &= \Big\{ -4 ; 8 \Big\} & &\end{align} \\ -----------------\)
  3. \((x+2)(5x-5)-x(4x-23)=-22\\ \Leftrightarrow 5x^2-5x+10x-10 -4x^2+23x+22=0 \\ \Leftrightarrow x^2+8x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.12 & &\\ & = 64-48 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-8-\sqrt16}{2.1} & & = \frac{-8+\sqrt16}{2.1} \\ & = \frac{-12}{2} & & = \frac{-4}{2} \\ & = -6 & & = -2 \\ \\ V &= \Big\{ -6 ; -2 \Big\} & &\end{align} \\ -----------------\)
  4. \(10x^2-(11x-121)=x(x+55) \\ \Leftrightarrow 10x^2-11x+121=x^2+55x \\ \Leftrightarrow 9x^2-66x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-66x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-66)^2-4.9.121 & &\\ & = 4356-4356 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-66)}{2.9} & & \\ & = \frac{11}{3} & & \\V &= \Big\{ \frac{11}{3} \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{1}{3}x^2+\frac{2}{3}x-21=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+\frac{2}{3}x-21\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2+2x-63=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-63=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-63) & &\\ & = 4+252 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt256}{2.1} & & = \frac{-2+\sqrt256}{2.1} \\ & = \frac{-18}{2} & & = \frac{14}{2} \\ & = -9 & & = 7 \\ \\ V &= \Big\{ -9 ; 7 \Big\} & &\end{align} \\ -----------------\)
  6. \(-(11-20x)=-72x^2-(9-13x) \\ \Leftrightarrow -11+20x=-72x^2-9+13x \\ \Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.72.(-2) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\ & = \frac{-32}{144} & & = \frac{18}{144} \\ & = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  7. \(-(9-22x)=-x^2-(45-10x) \\ \Leftrightarrow -9+22x=-x^2-45+10x \\ \Leftrightarrow x^2+12x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.36 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-12}{2.1} & & \\ & = -6 & & \\V &= \Big\{ -6 \Big\} & &\end{align} \\ -----------------\)
  8. \(-(3-17x)=-x^2-(52-3x) \\ \Leftrightarrow -3+17x=-x^2-52+3x \\ \Leftrightarrow x^2+14x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.49 & &\\ & = 196-196 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-14}{2.1} & & \\ & = -7 & & \\V &= \Big\{ -7 \Big\} & &\end{align} \\ -----------------\)
  9. \(2x^2-(10x-42)=x(x+3) \\ \Leftrightarrow 2x^2-10x+42=x^2+3x \\ \Leftrightarrow x^2-13x+42=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-13x+42=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-13)^2-4.1.42 & &\\ & = 169-168 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-13)-\sqrt1}{2.1} & & = \frac{-(-13)+\sqrt1}{2.1} \\ & = \frac{12}{2} & & = \frac{14}{2} \\ & = 6 & & = 7 \\ \\ V &= \Big\{ 6 ; 7 \Big\} & &\end{align} \\ -----------------\)
  10. \(\frac{5}{24}x=-\frac{1}{4}x^2+\frac{1}{4} \\ \Leftrightarrow \frac{1}{4}x^2+\frac{5}{24}x-\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{24.} \left(\frac{1}{4}x^2+\frac{5}{24}x-\frac{1}{4}\right)=0 \color{red}{.24} \\ \Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
  11. \((4x+4)(5x+4)-x(4x-48)=-84\\ \Leftrightarrow 20x^2+16x+20x+16 -4x^2+48x+84=0 \\ \Leftrightarrow 16x^2+80x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+80x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (80)^2-4.16.100 & &\\ & = 6400-6400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-80}{2.16} & & \\ & = -\frac{5}{2} & & \\V &= \Big\{ -\frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
  12. \(\frac{17}{4}x=-\frac{1}{2}x^2-2 \\ \Leftrightarrow \frac{1}{2}x^2+\frac{17}{4}x+2=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{2}x^2+\frac{17}{4}x+2\right)=0 \color{red}{.4} \\ \Leftrightarrow 2x^2+17x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+17x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.2.8 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.2} & & = \frac{-17+\sqrt225}{2.2} \\ & = \frac{-32}{4} & & = \frac{-2}{4} \\ & = -8 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -8 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-05-09 05:52:40
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