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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{25}{6}x=-8x^2-\frac{1}{2}\)
  2. \(\frac{1}{5}x^2+\frac{3}{4}x-\frac{1}{5}=0\)
  3. \(-(4-16x)=-4x^2-(5-11x)\)
  4. \((4x-3)(-2x+1)-x(-32x-6)=3\)
  5. \(7x^2-(4x+9)=3x(x-3)\)
  6. \(\frac{1}{5}x^2+\frac{13}{10}x+\frac{9}{5}=0\)
  7. \(\frac{3}{16}x^2+x+\frac{4}{3}=0\)
  8. \((4x-1)(-2x+5)-x(-9x+7)=43\)
  9. \(-(11-25x)=-x^2-(-1-14x)\)
  10. \(\frac{3}{2}x=-\frac{9}{16}x^2-1\)
  11. \(-(2-21x)=-x^2-(-42-14x)\)
  12. \(x(3x+19)=6(x-2)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{25}{6}x=-8x^2-\frac{1}{2} \\ \Leftrightarrow 8x^2+\frac{25}{6}x+\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{6.} \left(8x^2+\frac{25}{6}x+\frac{1}{2}\right)=0 \color{red}{.6} \\ \Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.48.3 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\ & = \frac{-32}{96} & & = \frac{-18}{96} \\ & = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
  2. \(\frac{1}{5}x^2+\frac{3}{4}x-\frac{1}{5}=0\\ \Leftrightarrow \color{red}{20.} \left(\frac{1}{5}x^2+\frac{3}{4}x-\frac{1}{5}\right)=0 \color{red}{.20} \\ \Leftrightarrow 4x^2+15x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.4.(-4) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\ & = \frac{-32}{8} & & = \frac{2}{8} \\ & = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
  3. \(-(4-16x)=-4x^2-(5-11x) \\ \Leftrightarrow -4+16x=-4x^2-5+11x \\ \Leftrightarrow 4x^2+5x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.1 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.4} & & = \frac{-5+\sqrt9}{2.4} \\ & = \frac{-8}{8} & & = \frac{-2}{8} \\ & = -1 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -1 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  4. \((4x-3)(-2x+1)-x(-32x-6)=3\\ \Leftrightarrow -8x^2+4x+6x-3 +32x^2+6x-3=0 \\ \Leftrightarrow 24x^2+7x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+7x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.24.(-6) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.24} & & = \frac{-7+\sqrt625}{2.24} \\ & = \frac{-32}{48} & & = \frac{18}{48} \\ & = \frac{-2}{3} & & = \frac{3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{3}{8} \Big\} & &\end{align} \\ -----------------\)
  5. \(7x^2-(4x+9)=3x(x-3) \\ \Leftrightarrow 7x^2-4x-9=3x^2-9x \\ \Leftrightarrow 4x^2+5x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.(-9) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.4} & & = \frac{-5+\sqrt169}{2.4} \\ & = \frac{-18}{8} & & = \frac{8}{8} \\ & = \frac{-9}{4} & & = 1 \\ \\ V &= \Big\{ \frac{-9}{4} ; 1 \Big\} & &\end{align} \\ -----------------\)
  6. \(\frac{1}{5}x^2+\frac{13}{10}x+\frac{9}{5}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{1}{5}x^2+\frac{13}{10}x+\frac{9}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 2x^2+13x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+13x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.2.18 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.2} & & = \frac{-13+\sqrt25}{2.2} \\ & = \frac{-18}{4} & & = \frac{-8}{4} \\ & = \frac{-9}{2} & & = -2 \\ \\ V &= \Big\{ \frac{-9}{2} ; -2 \Big\} & &\end{align} \\ -----------------\)
  7. \(\frac{3}{16}x^2+x+\frac{4}{3}=0\\ \Leftrightarrow \color{red}{48.} \left(\frac{3}{16}x^2+x+\frac{4}{3}\right)=0 \color{red}{.48} \\ \Leftrightarrow 9x^2+48x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+48x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (48)^2-4.9.64 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-48}{2.9} & & \\ & = -\frac{8}{3} & & \\V &= \Big\{ -\frac{8}{3} \Big\} & &\end{align} \\ -----------------\)
  8. \((4x-1)(-2x+5)-x(-9x+7)=43\\ \Leftrightarrow -8x^2+20x+2x-5 +9x^2-7x-43=0 \\ \Leftrightarrow x^2+8x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.(-48) & &\\ & = 64+192 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-8-\sqrt256}{2.1} & & = \frac{-8+\sqrt256}{2.1} \\ & = \frac{-24}{2} & & = \frac{8}{2} \\ & = -12 & & = 4 \\ \\ V &= \Big\{ -12 ; 4 \Big\} & &\end{align} \\ -----------------\)
  9. \(-(11-25x)=-x^2-(-1-14x) \\ \Leftrightarrow -11+25x=-x^2+1+14x \\ \Leftrightarrow x^2+11x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.(-12) & &\\ & = 121+48 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt169}{2.1} & & = \frac{-11+\sqrt169}{2.1} \\ & = \frac{-24}{2} & & = \frac{2}{2} \\ & = -12 & & = 1 \\ \\ V &= \Big\{ -12 ; 1 \Big\} & &\end{align} \\ -----------------\)
  10. \(\frac{3}{2}x=-\frac{9}{16}x^2-1 \\ \Leftrightarrow \frac{9}{16}x^2+\frac{3}{2}x+1=0 \\ \Leftrightarrow \color{red}{16.} \left(\frac{9}{16}x^2+\frac{3}{2}x+1\right)=0 \color{red}{.16} \\ \Leftrightarrow 9x^2+24x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+24x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.9.16 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.9} & & \\ & = -\frac{4}{3} & & \\V &= \Big\{ -\frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
  11. \(-(2-21x)=-x^2-(-42-14x) \\ \Leftrightarrow -2+21x=-x^2+42+14x \\ \Leftrightarrow x^2+7x-44=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+7x-44=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.(-44) & &\\ & = 49+176 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt225}{2.1} & & = \frac{-7+\sqrt225}{2.1} \\ & = \frac{-22}{2} & & = \frac{8}{2} \\ & = -11 & & = 4 \\ \\ V &= \Big\{ -11 ; 4 \Big\} & &\end{align} \\ -----------------\)
  12. \(x(3x+19)=6(x-2) \\ \Leftrightarrow 3x^2+19x=6x-12 \\ \Leftrightarrow 3x^2+13x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+13x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.3.12 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.3} & & = \frac{-13+\sqrt25}{2.3} \\ & = \frac{-18}{6} & & = \frac{-8}{6} \\ & = -3 & & = \frac{-4}{3} \\ \\ V &= \Big\{ -3 ; \frac{-4}{3} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-04-09 03:19:47
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