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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{4}{3}x^2+\frac{25}{6}x+3=0\)
  2. \(-(11-22x)=-9x^2-(15-9x)\)
  3. \(-(8-7x)=-9x^2-(4-2x)\)
  4. \(-(13-19x)=-x^2-(94-7x)\)
  5. \(8x^2-(7x-8)=6x(x-4)\)
  6. \(\frac{1}{4}x^2-\frac{3}{4}x-7=0\)
  7. \(\frac{2}{3}x=-\frac{3}{11}x^2-\frac{11}{3}\)
  8. \((-x+4)(5x+2)-x(-13x-1)=26\)
  9. \(-(4-15x)=-6x^2-(10-2x)\)
  10. \(-(13-9x)=-x^2-(-7-8x)\)
  11. \(-(5-6x)=-x^2-(15-13x)\)
  12. \(-\frac{8}{3}x=-\frac{1}{3}x^2-5\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{4}{3}x^2+\frac{25}{6}x+3=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{4}{3}x^2+\frac{25}{6}x+3\right)=0 \color{red}{.6} \\ \Leftrightarrow 8x^2+25x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+25x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.8.18 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.8} & & = \frac{-25+\sqrt49}{2.8} \\ & = \frac{-32}{16} & & = \frac{-18}{16} \\ & = -2 & & = \frac{-9}{8} \\ \\ V &= \Big\{ -2 ; \frac{-9}{8} \Big\} & &\end{align} \\ -----------------\)
  2. \(-(11-22x)=-9x^2-(15-9x) \\ \Leftrightarrow -11+22x=-9x^2-15+9x \\ \Leftrightarrow 9x^2+13x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+13x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.9.4 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.9} & & = \frac{-13+\sqrt25}{2.9} \\ & = \frac{-18}{18} & & = \frac{-8}{18} \\ & = -1 & & = \frac{-4}{9} \\ \\ V &= \Big\{ -1 ; \frac{-4}{9} \Big\} & &\end{align} \\ -----------------\)
  3. \(-(8-7x)=-9x^2-(4-2x) \\ \Leftrightarrow -8+7x=-9x^2-4+2x \\ \Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.9.(-4) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\ & = \frac{-18}{18} & & = \frac{8}{18} \\ & = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)
  4. \(-(13-19x)=-x^2-(94-7x) \\ \Leftrightarrow -13+19x=-x^2-94+7x \\ \Leftrightarrow x^2+12x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.81 & &\\ & = 144-324 & & \\ & = -180 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  5. \(8x^2-(7x-8)=6x(x-4) \\ \Leftrightarrow 8x^2-7x+8=6x^2-24x \\ \Leftrightarrow 2x^2+17x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+17x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.2.8 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.2} & & = \frac{-17+\sqrt225}{2.2} \\ & = \frac{-32}{4} & & = \frac{-2}{4} \\ & = -8 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -8 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
  6. \(\frac{1}{4}x^2-\frac{3}{4}x-7=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{3}{4}x-7\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-3x-28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-28=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-28) & &\\ & = 9+112 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt121}{2.1} & & = \frac{-(-3)+\sqrt121}{2.1} \\ & = \frac{-8}{2} & & = \frac{14}{2} \\ & = -4 & & = 7 \\ \\ V &= \Big\{ -4 ; 7 \Big\} & &\end{align} \\ -----------------\)
  7. \(\frac{2}{3}x=-\frac{3}{11}x^2-\frac{11}{3} \\ \Leftrightarrow \frac{3}{11}x^2+\frac{2}{3}x+\frac{11}{3}=0 \\ \Leftrightarrow \color{red}{33.} \left(\frac{3}{11}x^2+\frac{2}{3}x+\frac{11}{3}\right)=0 \color{red}{.33} \\ \Leftrightarrow 9x^2+22x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+22x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (22)^2-4.9.121 & &\\ & = 484-4356 & & \\ & = -3872 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  8. \((-x+4)(5x+2)-x(-13x-1)=26\\ \Leftrightarrow -5x^2-2x+20x+8 +13x^2+x-26=0 \\ \Leftrightarrow 8x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.8.(-18) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\ & = \frac{-32}{16} & & = \frac{18}{16} \\ & = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)
  9. \(-(4-15x)=-6x^2-(10-2x) \\ \Leftrightarrow -4+15x=-6x^2-10+2x \\ \Leftrightarrow 6x^2+13x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+13x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.6.6 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.6} & & = \frac{-13+\sqrt25}{2.6} \\ & = \frac{-18}{12} & & = \frac{-8}{12} \\ & = \frac{-3}{2} & & = \frac{-2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{-2}{3} \Big\} & &\end{align} \\ -----------------\)
  10. \(-(13-9x)=-x^2-(-7-8x) \\ \Leftrightarrow -13+9x=-x^2+7+8x \\ \Leftrightarrow x^2+x-20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-20) & &\\ & = 1+80 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt81}{2.1} & & = \frac{-1+\sqrt81}{2.1} \\ & = \frac{-10}{2} & & = \frac{8}{2} \\ & = -5 & & = 4 \\ \\ V &= \Big\{ -5 ; 4 \Big\} & &\end{align} \\ -----------------\)
  11. \(-(5-6x)=-x^2-(15-13x) \\ \Leftrightarrow -5+6x=-x^2-15+13x \\ \Leftrightarrow x^2-7x+10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x+10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.10 & &\\ & = 49-40 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt9}{2.1} & & = \frac{-(-7)+\sqrt9}{2.1} \\ & = \frac{4}{2} & & = \frac{10}{2} \\ & = 2 & & = 5 \\ \\ V &= \Big\{ 2 ; 5 \Big\} & &\end{align} \\ -----------------\)
  12. \(-\frac{8}{3}x=-\frac{1}{3}x^2-5 \\ \Leftrightarrow \frac{1}{3}x^2-\frac{8}{3}x+5=0 \\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2-\frac{8}{3}x+5\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2-8x+15=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+15=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.15 & &\\ & = 64-60 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt4}{2.1} & & = \frac{-(-8)+\sqrt4}{2.1} \\ & = \frac{6}{2} & & = \frac{10}{2} \\ & = 3 & & = 5 \\ \\ V &= \Big\{ 3 ; 5 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-01-27 23:24:39
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