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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \((3x+3)(-2x+2)-x(-12x-1)=0\)
  2. \(\frac{1}{4}x^2+\frac{5}{4}x-\frac{7}{2}=0\)
  3. \(-\frac{1}{2}x=-\frac{1}{18}x^2+2\)
  4. \(\frac{1}{9}x^2+x+\frac{9}{4}=0\)
  5. \(2x^2-\frac{17}{4}x+\frac{81}{8}=0\)
  6. \(7x^2-(10x+24)=x(x-17)\)
  7. \(\frac{9}{64}x^2-\frac{1}{4}x+1=0\)
  8. \((3x+5)(x-3)-x(2x-10)=-64\)
  9. \(\frac{1}{3}x^2+\frac{25}{36}x+\frac{1}{3}=0\)
  10. \(-(7-24x)=-x^2-(-53-20x)\)
  11. \(x(3x+33)=8(x-6)\)
  12. \(19x^2-(4x-8)=x(x-29)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \((3x+3)(-2x+2)-x(-12x-1)=0\\ \Leftrightarrow -6x^2+6x-6x+6 +12x^2+x+0=0 \\ \Leftrightarrow 6x^2+13x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+13x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.6.6 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.6} & & = \frac{-13+\sqrt25}{2.6} \\ & = \frac{-18}{12} & & = \frac{-8}{12} \\ & = \frac{-3}{2} & & = \frac{-2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{-2}{3} \Big\} & &\end{align} \\ -----------------\)
  2. \(\frac{1}{4}x^2+\frac{5}{4}x-\frac{7}{2}=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{5}{4}x-\frac{7}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+5x-14=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-14=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-14) & &\\ & = 25+56 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt81}{2.1} & & = \frac{-5+\sqrt81}{2.1} \\ & = \frac{-14}{2} & & = \frac{4}{2} \\ & = -7 & & = 2 \\ \\ V &= \Big\{ -7 ; 2 \Big\} & &\end{align} \\ -----------------\)
  3. \(-\frac{1}{2}x=-\frac{1}{18}x^2+2 \\ \Leftrightarrow \frac{1}{18}x^2-\frac{1}{2}x-2=0 \\ \Leftrightarrow \color{red}{18.} \left(\frac{1}{18}x^2-\frac{1}{2}x-2\right)=0 \color{red}{.18} \\ \Leftrightarrow x^2-9x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-9)^2-4.1.(-36) & &\\ & = 81+144 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-9)-\sqrt225}{2.1} & & = \frac{-(-9)+\sqrt225}{2.1} \\ & = \frac{-6}{2} & & = \frac{24}{2} \\ & = -3 & & = 12 \\ \\ V &= \Big\{ -3 ; 12 \Big\} & &\end{align} \\ -----------------\)
  4. \(\frac{1}{9}x^2+x+\frac{9}{4}=0\\ \Leftrightarrow \color{red}{36.} \left(\frac{1}{9}x^2+x+\frac{9}{4}\right)=0 \color{red}{.36} \\ \Leftrightarrow 4x^2+36x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+36x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (36)^2-4.4.81 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-36}{2.4} & & \\ & = -\frac{9}{2} & & \\V &= \Big\{ -\frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
  5. \(2x^2-\frac{17}{4}x+\frac{81}{8}=0\\ \Leftrightarrow \color{red}{8.} \left(2x^2-\frac{17}{4}x+\frac{81}{8}\right)=0 \color{red}{.8} \\ \Leftrightarrow 16x^2-34x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-34x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-34)^2-4.16.81 & &\\ & = 1156-5184 & & \\ & = -4028 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  6. \(7x^2-(10x+24)=x(x-17) \\ \Leftrightarrow 7x^2-10x-24=x^2-17x \\ \Leftrightarrow 6x^2+7x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+7x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.6.(-24) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.6} & & = \frac{-7+\sqrt625}{2.6} \\ & = \frac{-32}{12} & & = \frac{18}{12} \\ & = \frac{-8}{3} & & = \frac{3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
  7. \(\frac{9}{64}x^2-\frac{1}{4}x+1=0\\ \Leftrightarrow \color{red}{64.} \left(\frac{9}{64}x^2-\frac{1}{4}x+1\right)=0 \color{red}{.64} \\ \Leftrightarrow 9x^2-16x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-16x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.9.64 & &\\ & = 256-2304 & & \\ & = -2048 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  8. \((3x+5)(x-3)-x(2x-10)=-64\\ \Leftrightarrow 3x^2-9x+5x-15 -2x^2+10x+64=0 \\ \Leftrightarrow x^2-14x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.49 & &\\ & = 196-196 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-14)}{2.1} & & \\ & = 7 & & \\V &= \Big\{ 7 \Big\} & &\end{align} \\ -----------------\)
  9. \(\frac{1}{3}x^2+\frac{25}{36}x+\frac{1}{3}=0\\ \Leftrightarrow \color{red}{36.} \left(\frac{1}{3}x^2+\frac{25}{36}x+\frac{1}{3}\right)=0 \color{red}{.36} \\ \Leftrightarrow 12x^2+25x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+25x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.12.12 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.12} & & = \frac{-25+\sqrt49}{2.12} \\ & = \frac{-32}{24} & & = \frac{-18}{24} \\ & = \frac{-4}{3} & & = \frac{-3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{-3}{4} \Big\} & &\end{align} \\ -----------------\)
  10. \(-(7-24x)=-x^2-(-53-20x) \\ \Leftrightarrow -7+24x=-x^2+53+20x \\ \Leftrightarrow x^2+4x-60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-60) & &\\ & = 16+240 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt256}{2.1} & & = \frac{-4+\sqrt256}{2.1} \\ & = \frac{-20}{2} & & = \frac{12}{2} \\ & = -10 & & = 6 \\ \\ V &= \Big\{ -10 ; 6 \Big\} & &\end{align} \\ -----------------\)
  11. \(x(3x+33)=8(x-6) \\ \Leftrightarrow 3x^2+33x=8x-48 \\ \Leftrightarrow 3x^2+25x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+25x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.3.48 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.3} & & = \frac{-25+\sqrt49}{2.3} \\ & = \frac{-32}{6} & & = \frac{-18}{6} \\ & = \frac{-16}{3} & & = -3 \\ \\ V &= \Big\{ \frac{-16}{3} ; -3 \Big\} & &\end{align} \\ -----------------\)
  12. \(19x^2-(4x-8)=x(x-29) \\ \Leftrightarrow 19x^2-4x+8=x^2-29x \\ \Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-04-01 18:17:19
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