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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{1}{3}x^2+\frac{5}{3}x+\frac{4}{3}=0\)
  2. \(\frac{7}{12}x=-\frac{1}{4}x^2+4\)
  3. \((5x-3)(3x-3)-x(3x-13)=21\)
  4. \((x-5)(-x-5)-x(-17x+3)=24\)
  5. \(4x^2+\frac{5}{2}x+\frac{1}{4}=0\)
  6. \(5x^2-(9x-121)=x(x+3)\)
  7. \((-5x-5)(-2x+2)-x(9x-22)=25\)
  8. \(\frac{1}{6}x^2+\frac{1}{3}x+24=0\)
  9. \(-\frac{3}{2}x=-\frac{9}{40}x^2-\frac{5}{2}\)
  10. \((3x-3)(-3x-5)-x(-10x-3)=85\)
  11. \((-x+5)(-x+4)-x(-3x+28)=-5\)
  12. \(-(4-59x)=-16x^2-(68-13x)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{1}{3}x^2+\frac{5}{3}x+\frac{4}{3}=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+\frac{5}{3}x+\frac{4}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2+5x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.4 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.1} & & = \frac{-5+\sqrt9}{2.1} \\ & = \frac{-8}{2} & & = \frac{-2}{2} \\ & = -4 & & = -1 \\ \\ V &= \Big\{ -4 ; -1 \Big\} & &\end{align} \\ -----------------\)
  2. \(\frac{7}{12}x=-\frac{1}{4}x^2+4 \\ \Leftrightarrow \frac{1}{4}x^2+\frac{7}{12}x-4=0 \\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{4}x^2+\frac{7}{12}x-4\right)=0 \color{red}{.12} \\ \Leftrightarrow 3x^2+7x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+7x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.3.(-48) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.3} & & = \frac{-7+\sqrt625}{2.3} \\ & = \frac{-32}{6} & & = \frac{18}{6} \\ & = \frac{-16}{3} & & = 3 \\ \\ V &= \Big\{ \frac{-16}{3} ; 3 \Big\} & &\end{align} \\ -----------------\)
  3. \((5x-3)(3x-3)-x(3x-13)=21\\ \Leftrightarrow 15x^2-15x-9x+9 -3x^2+13x-21=0 \\ \Leftrightarrow 12x^2+7x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+7x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.12.(-12) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.12} & & = \frac{-7+\sqrt625}{2.12} \\ & = \frac{-32}{24} & & = \frac{18}{24} \\ & = \frac{-4}{3} & & = \frac{3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
  4. \((x-5)(-x-5)-x(-17x+3)=24\\ \Leftrightarrow -x^2-5x+5x+25 +17x^2-3x-24=0 \\ \Leftrightarrow 16x^2+17x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+17x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.16.1 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.16} & & = \frac{-17+\sqrt225}{2.16} \\ & = \frac{-32}{32} & & = \frac{-2}{32} \\ & = -1 & & = \frac{-1}{16} \\ \\ V &= \Big\{ -1 ; \frac{-1}{16} \Big\} & &\end{align} \\ -----------------\)
  5. \(4x^2+\frac{5}{2}x+\frac{1}{4}=0\\ \Leftrightarrow \color{red}{4.} \left(4x^2+\frac{5}{2}x+\frac{1}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow 16x^2+10x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+10x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.16.1 & &\\ & = 100-64 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt36}{2.16} & & = \frac{-10+\sqrt36}{2.16} \\ & = \frac{-16}{32} & & = \frac{-4}{32} \\ & = \frac{-1}{2} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
  6. \(5x^2-(9x-121)=x(x+3) \\ \Leftrightarrow 5x^2-9x+121=x^2+3x \\ \Leftrightarrow 4x^2-12x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-12x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.4.121 & &\\ & = 144-1936 & & \\ & = -1792 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  7. \((-5x-5)(-2x+2)-x(9x-22)=25\\ \Leftrightarrow 10x^2-10x+10x-10 -9x^2+22x-25=0 \\ \Leftrightarrow x^2+2x-35=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-35=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-35) & &\\ & = 4+140 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt144}{2.1} & & = \frac{-2+\sqrt144}{2.1} \\ & = \frac{-14}{2} & & = \frac{10}{2} \\ & = -7 & & = 5 \\ \\ V &= \Big\{ -7 ; 5 \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{1}{6}x^2+\frac{1}{3}x+24=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{6}x^2+\frac{1}{3}x+24\right)=0 \color{red}{.6} \\ \Leftrightarrow x^2+2x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.144 & &\\ & = 4-576 & & \\ & = -572 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  9. \(-\frac{3}{2}x=-\frac{9}{40}x^2-\frac{5}{2} \\ \Leftrightarrow \frac{9}{40}x^2-\frac{3}{2}x+\frac{5}{2}=0 \\ \Leftrightarrow \color{red}{40.} \left(\frac{9}{40}x^2-\frac{3}{2}x+\frac{5}{2}\right)=0 \color{red}{.40} \\ \Leftrightarrow 9x^2-60x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-60x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-60)^2-4.9.100 & &\\ & = 3600-3600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-60)}{2.9} & & \\ & = \frac{10}{3} & & \\V &= \Big\{ \frac{10}{3} \Big\} & &\end{align} \\ -----------------\)
  10. \((3x-3)(-3x-5)-x(-10x-3)=85\\ \Leftrightarrow -9x^2-15x+9x+15 +10x^2+3x-85=0 \\ \Leftrightarrow x^2+3x-70=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-70=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-70) & &\\ & = 9+280 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt289}{2.1} & & = \frac{-3+\sqrt289}{2.1} \\ & = \frac{-20}{2} & & = \frac{14}{2} \\ & = -10 & & = 7 \\ \\ V &= \Big\{ -10 ; 7 \Big\} & &\end{align} \\ -----------------\)
  11. \((-x+5)(-x+4)-x(-3x+28)=-5\\ \Leftrightarrow x^2-4x-5x+20 +3x^2-28x+5=0 \\ \Leftrightarrow 4x^2-12x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-12x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.4.25 & &\\ & = 144-400 & & \\ & = -256 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  12. \(-(4-59x)=-16x^2-(68-13x) \\ \Leftrightarrow -4+59x=-16x^2-68+13x \\ \Leftrightarrow 16x^2+46x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+46x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (46)^2-4.16.64 & &\\ & = 2116-4096 & & \\ & = -1980 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-05-05 16:29:38
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