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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(x(9x-33)=-25(x+1)\)
  2. \(\frac{9}{5}x^2+\frac{13}{5}x+\frac{4}{5}=0\)
  3. \(\frac{1}{4}x=-\frac{1}{8}x^2+1\)
  4. \((-2x-4)(5x+2)-x(-14x-17)=1\)
  5. \(\frac{1}{2}x^2+\frac{13}{6}x+2=0\)
  6. \(10x^2-(3x-64)=x(x-17)\)
  7. \(2x^2-(12x-120)=x(x+10)\)
  8. \(-(9-23x)=-2x^2-(-63-16x)\)
  9. \((x+1)(-2x+1)-x(-3x+5)=89\)
  10. \(17x^2-(12x-3)=5x(x-5)\)
  11. \(10x^2-(8x-144)=x(x+64)\)
  12. \(\frac{1}{4}x^2-\frac{17}{4}x+\frac{35}{2}=0\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(x(9x-33)=-25(x+1) \\ \Leftrightarrow 9x^2-33x=-25x-25 \\ \Leftrightarrow 9x^2-8x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-8x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.9.25 & &\\ & = 64-900 & & \\ & = -836 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  2. \(\frac{9}{5}x^2+\frac{13}{5}x+\frac{4}{5}=0\\ \Leftrightarrow \color{red}{5.} \left(\frac{9}{5}x^2+\frac{13}{5}x+\frac{4}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow 9x^2+13x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+13x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.9.4 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.9} & & = \frac{-13+\sqrt25}{2.9} \\ & = \frac{-18}{18} & & = \frac{-8}{18} \\ & = -1 & & = \frac{-4}{9} \\ \\ V &= \Big\{ -1 ; \frac{-4}{9} \Big\} & &\end{align} \\ -----------------\)
  3. \(\frac{1}{4}x=-\frac{1}{8}x^2+1 \\ \Leftrightarrow \frac{1}{8}x^2+\frac{1}{4}x-1=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2+\frac{1}{4}x-1\right)=0 \color{red}{.8} \\ \Leftrightarrow x^2+2x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-8) & &\\ & = 4+32 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt36}{2.1} & & = \frac{-2+\sqrt36}{2.1} \\ & = \frac{-8}{2} & & = \frac{4}{2} \\ & = -4 & & = 2 \\ \\ V &= \Big\{ -4 ; 2 \Big\} & &\end{align} \\ -----------------\)
  4. \((-2x-4)(5x+2)-x(-14x-17)=1\\ \Leftrightarrow -10x^2-4x-20x-8 +14x^2+17x-1=0 \\ \Leftrightarrow 4x^2+5x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.(-9) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.4} & & = \frac{-5+\sqrt169}{2.4} \\ & = \frac{-18}{8} & & = \frac{8}{8} \\ & = \frac{-9}{4} & & = 1 \\ \\ V &= \Big\{ \frac{-9}{4} ; 1 \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{1}{2}x^2+\frac{13}{6}x+2=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{2}x^2+\frac{13}{6}x+2\right)=0 \color{red}{.6} \\ \Leftrightarrow 3x^2+13x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+13x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.3.12 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.3} & & = \frac{-13+\sqrt25}{2.3} \\ & = \frac{-18}{6} & & = \frac{-8}{6} \\ & = -3 & & = \frac{-4}{3} \\ \\ V &= \Big\{ -3 ; \frac{-4}{3} \Big\} & &\end{align} \\ -----------------\)
  6. \(10x^2-(3x-64)=x(x-17) \\ \Leftrightarrow 10x^2-3x+64=x^2-17x \\ \Leftrightarrow 9x^2+14x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+14x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.9.64 & &\\ & = 196-2304 & & \\ & = -2108 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  7. \(2x^2-(12x-120)=x(x+10) \\ \Leftrightarrow 2x^2-12x+120=x^2+10x \\ \Leftrightarrow x^2-22x+120=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-22x+120=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.1.120 & &\\ & = 484-480 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-22)-\sqrt4}{2.1} & & = \frac{-(-22)+\sqrt4}{2.1} \\ & = \frac{20}{2} & & = \frac{24}{2} \\ & = 10 & & = 12 \\ \\ V &= \Big\{ 10 ; 12 \Big\} & &\end{align} \\ -----------------\)
  8. \(-(9-23x)=-2x^2-(-63-16x) \\ \Leftrightarrow -9+23x=-2x^2+63+16x \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
  9. \((x+1)(-2x+1)-x(-3x+5)=89\\ \Leftrightarrow -2x^2+x-2x+1 +3x^2-5x-89=0 \\ \Leftrightarrow x^2-3x-88=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-88=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-88) & &\\ & = 9+352 & & \\ & = 361 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt361}{2.1} & & = \frac{-(-3)+\sqrt361}{2.1} \\ & = \frac{-16}{2} & & = \frac{22}{2} \\ & = -8 & & = 11 \\ \\ V &= \Big\{ -8 ; 11 \Big\} & &\end{align} \\ -----------------\)
  10. \(17x^2-(12x-3)=5x(x-5) \\ \Leftrightarrow 17x^2-12x+3=5x^2-25x \\ \Leftrightarrow 12x^2+13x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+13x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.12.3 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.12} & & = \frac{-13+\sqrt25}{2.12} \\ & = \frac{-18}{24} & & = \frac{-8}{24} \\ & = \frac{-3}{4} & & = \frac{-1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{-1}{3} \Big\} & &\end{align} \\ -----------------\)
  11. \(10x^2-(8x-144)=x(x+64) \\ \Leftrightarrow 10x^2-8x+144=x^2+64x \\ \Leftrightarrow 9x^2-72x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-72x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-72)^2-4.9.144 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-72)}{2.9} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
  12. \(\frac{1}{4}x^2-\frac{17}{4}x+\frac{35}{2}=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{17}{4}x+\frac{35}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-17x+70=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-17x+70=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-17)^2-4.1.70 & &\\ & = 289-280 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-17)-\sqrt9}{2.1} & & = \frac{-(-17)+\sqrt9}{2.1} \\ & = \frac{14}{2} & & = \frac{20}{2} \\ & = 7 & & = 10 \\ \\ V &= \Big\{ 7 ; 10 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator vanhoeckes.be/wiskunde 2025-05-30 04:41:55