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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{9}{16}x^2-\frac{3}{2}x+1=0\)
  2. \(-(13-7x)=-x^2-(17-11x)\)
  3. \(7x^2-(15x-24)=x(x-40)\)
  4. \(4x^2-(2x+48)=x(x-9)\)
  5. \(-(10+19x)=-9x^2-(35-5x)\)
  6. \(-(5-36x)=-36x^2-(9-11x)\)
  7. \((-2x+4)(-x-5)-x(-46x-35)=-23\)
  8. \((-4x-5)(-5x-2)-x(14x+13)=16\)
  9. \(-\frac{1}{4}x=-\frac{1}{48}x^2-\frac{3}{4}\)
  10. \(\frac{1}{12}x^2-x+\frac{9}{4}=0\)
  11. \((4x+1)(5x-1)-x(16x-22)=-5\)
  12. \(x(6x+31)=24(x+1)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{9}{16}x^2-\frac{3}{2}x+1=0\\ \Leftrightarrow \color{red}{16.} \left(\frac{9}{16}x^2-\frac{3}{2}x+1\right)=0 \color{red}{.16} \\ \Leftrightarrow 9x^2-24x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-24x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.9.16 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-24)}{2.9} & & \\ & = \frac{4}{3} & & \\V &= \Big\{ \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
  2. \(-(13-7x)=-x^2-(17-11x) \\ \Leftrightarrow -13+7x=-x^2-17+11x \\ \Leftrightarrow x^2-4x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-4)}{2.1} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
  3. \(7x^2-(15x-24)=x(x-40) \\ \Leftrightarrow 7x^2-15x+24=x^2-40x \\ \Leftrightarrow 6x^2+25x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+25x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.6.24 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.6} & & = \frac{-25+\sqrt49}{2.6} \\ & = \frac{-32}{12} & & = \frac{-18}{12} \\ & = \frac{-8}{3} & & = \frac{-3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{-3}{2} \Big\} & &\end{align} \\ -----------------\)
  4. \(4x^2-(2x+48)=x(x-9) \\ \Leftrightarrow 4x^2-2x-48=x^2-9x \\ \Leftrightarrow 3x^2+7x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+7x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.3.(-48) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.3} & & = \frac{-7+\sqrt625}{2.3} \\ & = \frac{-32}{6} & & = \frac{18}{6} \\ & = \frac{-16}{3} & & = 3 \\ \\ V &= \Big\{ \frac{-16}{3} ; 3 \Big\} & &\end{align} \\ -----------------\)
  5. \(-(10+19x)=-9x^2-(35-5x) \\ \Leftrightarrow -10-19x=-9x^2-35+5x \\ \Leftrightarrow 9x^2-24x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-24x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.9.25 & &\\ & = 576-900 & & \\ & = -324 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  6. \(-(5-36x)=-36x^2-(9-11x) \\ \Leftrightarrow -5+36x=-36x^2-9+11x \\ \Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  7. \((-2x+4)(-x-5)-x(-46x-35)=-23\\ \Leftrightarrow 2x^2+10x-4x-20 +46x^2+35x+23=0 \\ \Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.48.3 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\ & = \frac{-32}{96} & & = \frac{-18}{96} \\ & = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
  8. \((-4x-5)(-5x-2)-x(14x+13)=16\\ \Leftrightarrow 20x^2+8x+25x+10 -14x^2-13x-16=0 \\ \Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
  9. \(-\frac{1}{4}x=-\frac{1}{48}x^2-\frac{3}{4} \\ \Leftrightarrow \frac{1}{48}x^2-\frac{1}{4}x+\frac{3}{4}=0 \\ \Leftrightarrow \color{red}{48.} \left(\frac{1}{48}x^2-\frac{1}{4}x+\frac{3}{4}\right)=0 \color{red}{.48} \\ \Leftrightarrow 4x^2-48x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-48x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-48)^2-4.4.144 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-48)}{2.4} & & \\ & = 6 & & \\V &= \Big\{ 6 \Big\} & &\end{align} \\ -----------------\)
  10. \(\frac{1}{12}x^2-x+\frac{9}{4}=0\\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{12}x^2-x+\frac{9}{4}\right)=0 \color{red}{.12} \\ \Leftrightarrow x^2-12x+27=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+27=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.27 & &\\ & = 144-108 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-12)-\sqrt36}{2.1} & & = \frac{-(-12)+\sqrt36}{2.1} \\ & = \frac{6}{2} & & = \frac{18}{2} \\ & = 3 & & = 9 \\ \\ V &= \Big\{ 3 ; 9 \Big\} & &\end{align} \\ -----------------\)
  11. \((4x+1)(5x-1)-x(16x-22)=-5\\ \Leftrightarrow 20x^2-4x+5x-1 -16x^2+22x+5=0 \\ \Leftrightarrow 4x^2+17x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+17x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.4.4 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.4} & & = \frac{-17+\sqrt225}{2.4} \\ & = \frac{-32}{8} & & = \frac{-2}{8} \\ & = -4 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -4 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  12. \(x(6x+31)=24(x+1) \\ \Leftrightarrow 6x^2+31x=24x+24 \\ \Leftrightarrow 6x^2+7x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+7x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.6.(-24) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.6} & & = \frac{-7+\sqrt625}{2.6} \\ & = \frac{-32}{12} & & = \frac{18}{12} \\ & = \frac{-8}{3} & & = \frac{3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-02-19 23:27:49
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