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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(2x^2-(4x+35)=x(x-6)\)
  2. \(-3x=-\frac{1}{2}x^2-\frac{25}{2}\)
  3. \((-3x-2)(4x+2)-x(-13x-26)=-68\)
  4. \(-\frac{1}{4}x=-\frac{1}{8}x^2+15\)
  5. \(\frac{1}{3}x^2+\frac{25}{6}x+12=0\)
  6. \(x(x-9)=8(x-9)\)
  7. \((4x-2)(x+1)-x(3x+7)=-8\)
  8. \(2x^2-(11x-8)=x(x-17)\)
  9. \(\frac{1}{2}x=-\frac{1}{2}x^2+3\)
  10. \((-x-3)(-4x-3)-x(0x-1)=0\)
  11. \((-4x-5)(2x+1)-x(-17x-14)=-1\)
  12. \(7x^2-(8x+6)=x(x-13)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(2x^2-(4x+35)=x(x-6) \\ \Leftrightarrow 2x^2-4x-35=x^2-6x \\ \Leftrightarrow x^2+2x-35=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-35=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-35) & &\\ & = 4+140 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt144}{2.1} & & = \frac{-2+\sqrt144}{2.1} \\ & = \frac{-14}{2} & & = \frac{10}{2} \\ & = -7 & & = 5 \\ \\ V &= \Big\{ -7 ; 5 \Big\} & &\end{align} \\ -----------------\)
  2. \(-3x=-\frac{1}{2}x^2-\frac{25}{2} \\ \Leftrightarrow \frac{1}{2}x^2-3x+\frac{25}{2}=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-3x+\frac{25}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2-6x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.25 & &\\ & = 36-100 & & \\ & = -64 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  3. \((-3x-2)(4x+2)-x(-13x-26)=-68\\ \Leftrightarrow -12x^2-6x-8x-4 +13x^2+26x+68=0 \\ \Leftrightarrow x^2+16x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.64 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.1} & & \\ & = -8 & & \\V &= \Big\{ -8 \Big\} & &\end{align} \\ -----------------\)
  4. \(-\frac{1}{4}x=-\frac{1}{8}x^2+15 \\ \Leftrightarrow \frac{1}{8}x^2-\frac{1}{4}x-15=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2-\frac{1}{4}x-15\right)=0 \color{red}{.8} \\ \Leftrightarrow x^2-2x-120=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-120=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-120) & &\\ & = 4+480 & & \\ & = 484 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt484}{2.1} & & = \frac{-(-2)+\sqrt484}{2.1} \\ & = \frac{-20}{2} & & = \frac{24}{2} \\ & = -10 & & = 12 \\ \\ V &= \Big\{ -10 ; 12 \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{1}{3}x^2+\frac{25}{6}x+12=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{3}x^2+\frac{25}{6}x+12\right)=0 \color{red}{.6} \\ \Leftrightarrow 2x^2+25x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+25x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.2.72 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.2} & & = \frac{-25+\sqrt49}{2.2} \\ & = \frac{-32}{4} & & = \frac{-18}{4} \\ & = -8 & & = \frac{-9}{2} \\ \\ V &= \Big\{ -8 ; \frac{-9}{2} \Big\} & &\end{align} \\ -----------------\)
  6. \(x(x-9)=8(x-9) \\ \Leftrightarrow x^2-9x=8x-72 \\ \Leftrightarrow x^2-17x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-17x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-17)^2-4.1.72 & &\\ & = 289-288 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-17)-\sqrt1}{2.1} & & = \frac{-(-17)+\sqrt1}{2.1} \\ & = \frac{16}{2} & & = \frac{18}{2} \\ & = 8 & & = 9 \\ \\ V &= \Big\{ 8 ; 9 \Big\} & &\end{align} \\ -----------------\)
  7. \((4x-2)(x+1)-x(3x+7)=-8\\ \Leftrightarrow 4x^2+4x-2x-2 -3x^2-7x+8=0 \\ \Leftrightarrow x^2-5x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.6 & &\\ & = 25-24 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt1}{2.1} & & = \frac{-(-5)+\sqrt1}{2.1} \\ & = \frac{4}{2} & & = \frac{6}{2} \\ & = 2 & & = 3 \\ \\ V &= \Big\{ 2 ; 3 \Big\} & &\end{align} \\ -----------------\)
  8. \(2x^2-(11x-8)=x(x-17) \\ \Leftrightarrow 2x^2-11x+8=x^2-17x \\ \Leftrightarrow x^2+6x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.8 & &\\ & = 36-32 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt4}{2.1} & & = \frac{-6+\sqrt4}{2.1} \\ & = \frac{-8}{2} & & = \frac{-4}{2} \\ & = -4 & & = -2 \\ \\ V &= \Big\{ -4 ; -2 \Big\} & &\end{align} \\ -----------------\)
  9. \(\frac{1}{2}x=-\frac{1}{2}x^2+3 \\ \Leftrightarrow \frac{1}{2}x^2+\frac{1}{2}x-3=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+\frac{1}{2}x-3\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2+x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-6) & &\\ & = 1+24 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt25}{2.1} & & = \frac{-1+\sqrt25}{2.1} \\ & = \frac{-6}{2} & & = \frac{4}{2} \\ & = -3 & & = 2 \\ \\ V &= \Big\{ -3 ; 2 \Big\} & &\end{align} \\ -----------------\)
  10. \((-x-3)(-4x-3)-x(0x-1)=0\\ \Leftrightarrow 4x^2+3x+12x+9 +0x^2+x+0=0 \\ \Leftrightarrow 4x^2+13x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+13x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.4.9 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.4} & & = \frac{-13+\sqrt25}{2.4} \\ & = \frac{-18}{8} & & = \frac{-8}{8} \\ & = \frac{-9}{4} & & = -1 \\ \\ V &= \Big\{ \frac{-9}{4} ; -1 \Big\} & &\end{align} \\ -----------------\)
  11. \((-4x-5)(2x+1)-x(-17x-14)=-1\\ \Leftrightarrow -8x^2-4x-10x-5 +17x^2+14x+1=0 \\ \Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.9.(-4) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\ & = \frac{-18}{18} & & = \frac{8}{18} \\ & = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)
  12. \(7x^2-(8x+6)=x(x-13) \\ \Leftrightarrow 7x^2-8x-6=x^2-13x \\ \Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-09-29 09:18:19
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