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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{1}{32}x^2-\frac{1}{4}x+\frac{1}{2}=0\)
  2. \((x+3)(-5x-3)-x(-6x+0)=-41\)
  3. \(\frac{1}{8}x^2+\frac{3}{2}x+4=0\)
  4. \(\frac{1}{3}x^2+\frac{11}{3}x+\frac{28}{3}=0\)
  5. \(58x^2-(15x-3)=10x(x-4)\)
  6. \((-3x+3)(3x-1)-x(-17x-15)=-1\)
  7. \((x+4)(-x-2)-x(-10x-22)=-12\)
  8. \(\frac{1}{5}x^2+\frac{7}{20}x-\frac{9}{5}=0\)
  9. \((4x+4)(-3x-5)-x(-13x-32)=-11\)
  10. \(-(6-13x)=-9x^2-(10-19x)\)
  11. \(2x^2+\frac{7}{4}x-\frac{9}{2}=0\)
  12. \(\frac{4}{3}x^2+\frac{13}{3}x+3=0\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{1}{32}x^2-\frac{1}{4}x+\frac{1}{2}=0\\ \Leftrightarrow \color{red}{32.} \left(\frac{1}{32}x^2-\frac{1}{4}x+\frac{1}{2}\right)=0 \color{red}{.32} \\ \Leftrightarrow x^2-8x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.16 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-8)}{2.1} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
  2. \((x+3)(-5x-3)-x(-6x+0)=-41\\ \Leftrightarrow -5x^2-3x-15x-9 +6x^2+0x+41=0 \\ \Leftrightarrow x^2-12x+32=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+32=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.32 & &\\ & = 144-128 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-12)-\sqrt16}{2.1} & & = \frac{-(-12)+\sqrt16}{2.1} \\ & = \frac{8}{2} & & = \frac{16}{2} \\ & = 4 & & = 8 \\ \\ V &= \Big\{ 4 ; 8 \Big\} & &\end{align} \\ -----------------\)
  3. \(\frac{1}{8}x^2+\frac{3}{2}x+4=0\\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2+\frac{3}{2}x+4\right)=0 \color{red}{.8} \\ \Leftrightarrow x^2+12x+32=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+32=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.32 & &\\ & = 144-128 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-12-\sqrt16}{2.1} & & = \frac{-12+\sqrt16}{2.1} \\ & = \frac{-16}{2} & & = \frac{-8}{2} \\ & = -8 & & = -4 \\ \\ V &= \Big\{ -8 ; -4 \Big\} & &\end{align} \\ -----------------\)
  4. \(\frac{1}{3}x^2+\frac{11}{3}x+\frac{28}{3}=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+\frac{11}{3}x+\frac{28}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2+11x+28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+28=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.28 & &\\ & = 121-112 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt9}{2.1} & & = \frac{-11+\sqrt9}{2.1} \\ & = \frac{-14}{2} & & = \frac{-8}{2} \\ & = -7 & & = -4 \\ \\ V &= \Big\{ -7 ; -4 \Big\} & &\end{align} \\ -----------------\)
  5. \(58x^2-(15x-3)=10x(x-4) \\ \Leftrightarrow 58x^2-15x+3=10x^2-40x \\ \Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.48.3 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\ & = \frac{-32}{96} & & = \frac{-18}{96} \\ & = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
  6. \((-3x+3)(3x-1)-x(-17x-15)=-1\\ \Leftrightarrow -9x^2+3x+9x-3 +17x^2+15x+1=0 \\ \Leftrightarrow 8x^2+15x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+15x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.8.(-2) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.8} & & = \frac{-15+\sqrt289}{2.8} \\ & = \frac{-32}{16} & & = \frac{2}{16} \\ & = -2 & & = \frac{1}{8} \\ \\ V &= \Big\{ -2 ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  7. \((x+4)(-x-2)-x(-10x-22)=-12\\ \Leftrightarrow -x^2-2x-4x-8 +10x^2+22x+12=0 \\ \Leftrightarrow 9x^2+12x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+12x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.9.4 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-12}{2.9} & & \\ & = -\frac{2}{3} & & \\V &= \Big\{ -\frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{1}{5}x^2+\frac{7}{20}x-\frac{9}{5}=0\\ \Leftrightarrow \color{red}{20.} \left(\frac{1}{5}x^2+\frac{7}{20}x-\frac{9}{5}\right)=0 \color{red}{.20} \\ \Leftrightarrow 4x^2+7x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+7x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.4.(-36) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.4} & & = \frac{-7+\sqrt625}{2.4} \\ & = \frac{-32}{8} & & = \frac{18}{8} \\ & = -4 & & = \frac{9}{4} \\ \\ V &= \Big\{ -4 ; \frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
  9. \((4x+4)(-3x-5)-x(-13x-32)=-11\\ \Leftrightarrow -12x^2-20x-12x-20 +13x^2+32x+11=0 \\ \Leftrightarrow x^2-8x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-9) & &\\ & = 64+36 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt100}{2.1} & & = \frac{-(-8)+\sqrt100}{2.1} \\ & = \frac{-2}{2} & & = \frac{18}{2} \\ & = -1 & & = 9 \\ \\ V &= \Big\{ -1 ; 9 \Big\} & &\end{align} \\ -----------------\)
  10. \(-(6-13x)=-9x^2-(10-19x) \\ \Leftrightarrow -6+13x=-9x^2-10+19x \\ \Leftrightarrow 9x^2-6x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-6x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.9.4 & &\\ & = 36-144 & & \\ & = -108 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  11. \(2x^2+\frac{7}{4}x-\frac{9}{2}=0\\ \Leftrightarrow \color{red}{4.} \left(2x^2+\frac{7}{4}x-\frac{9}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow 8x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.8.(-18) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\ & = \frac{-32}{16} & & = \frac{18}{16} \\ & = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)
  12. \(\frac{4}{3}x^2+\frac{13}{3}x+3=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{4}{3}x^2+\frac{13}{3}x+3\right)=0 \color{red}{.3} \\ \Leftrightarrow 4x^2+13x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+13x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.4.9 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.4} & & = \frac{-13+\sqrt25}{2.4} \\ & = \frac{-18}{8} & & = \frac{-8}{8} \\ & = \frac{-9}{4} & & = -1 \\ \\ V &= \Big\{ \frac{-9}{4} ; -1 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-06-22 16:25:37
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