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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-(3-38x)=-8x^2-(21-13x)\)
  2. \(x(8x+17)=2(x+1)\)
  3. \(\frac{36}{5}x^2+\frac{7}{10}x-\frac{1}{5}=0\)
  4. \(x(12x+10)=-3(x+1)\)
  5. \(-(14-14x)=-x^2-(-36-9x)\)
  6. \(\frac{1}{5}x^2+\frac{1}{5}x-\frac{72}{5}=0\)
  7. \(\frac{1}{4}x^2+\frac{25}{24}x+1=0\)
  8. \(x(x+46)=40(x+1)\)
  9. \((2x+3)(-x-3)-x(-3x+1)=-73\)
  10. \((4x+2)(4x-3)-x(15x-34)=-70\)
  11. \(\frac{1}{2}x^2-\frac{3}{4}x+\frac{81}{32}=0\)
  12. \(\frac{1}{10}x^2-\frac{1}{5}x-\frac{24}{5}=0\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-(3-38x)=-8x^2-(21-13x) \\ \Leftrightarrow -3+38x=-8x^2-21+13x \\ \Leftrightarrow 8x^2+25x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+25x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.8.18 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.8} & & = \frac{-25+\sqrt49}{2.8} \\ & = \frac{-32}{16} & & = \frac{-18}{16} \\ & = -2 & & = \frac{-9}{8} \\ \\ V &= \Big\{ -2 ; \frac{-9}{8} \Big\} & &\end{align} \\ -----------------\)
  2. \(x(8x+17)=2(x+1) \\ \Leftrightarrow 8x^2+17x=2x+2 \\ \Leftrightarrow 8x^2+15x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+15x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.8.(-2) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.8} & & = \frac{-15+\sqrt289}{2.8} \\ & = \frac{-32}{16} & & = \frac{2}{16} \\ & = -2 & & = \frac{1}{8} \\ \\ V &= \Big\{ -2 ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  3. \(\frac{36}{5}x^2+\frac{7}{10}x-\frac{1}{5}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{36}{5}x^2+\frac{7}{10}x-\frac{1}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.72.(-2) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\ & = \frac{-32}{144} & & = \frac{18}{144} \\ & = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  4. \(x(12x+10)=-3(x+1) \\ \Leftrightarrow 12x^2+10x=-3x-3 \\ \Leftrightarrow 12x^2+13x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+13x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.12.3 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.12} & & = \frac{-13+\sqrt25}{2.12} \\ & = \frac{-18}{24} & & = \frac{-8}{24} \\ & = \frac{-3}{4} & & = \frac{-1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{-1}{3} \Big\} & &\end{align} \\ -----------------\)
  5. \(-(14-14x)=-x^2-(-36-9x) \\ \Leftrightarrow -14+14x=-x^2+36+9x \\ \Leftrightarrow x^2+5x-50=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-50=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-50) & &\\ & = 25+200 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt225}{2.1} & & = \frac{-5+\sqrt225}{2.1} \\ & = \frac{-20}{2} & & = \frac{10}{2} \\ & = -10 & & = 5 \\ \\ V &= \Big\{ -10 ; 5 \Big\} & &\end{align} \\ -----------------\)
  6. \(\frac{1}{5}x^2+\frac{1}{5}x-\frac{72}{5}=0\\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+\frac{1}{5}x-\frac{72}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2+x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-72) & &\\ & = 1+288 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt289}{2.1} & & = \frac{-1+\sqrt289}{2.1} \\ & = \frac{-18}{2} & & = \frac{16}{2} \\ & = -9 & & = 8 \\ \\ V &= \Big\{ -9 ; 8 \Big\} & &\end{align} \\ -----------------\)
  7. \(\frac{1}{4}x^2+\frac{25}{24}x+1=0\\ \Leftrightarrow \color{red}{24.} \left(\frac{1}{4}x^2+\frac{25}{24}x+1\right)=0 \color{red}{.24} \\ \Leftrightarrow 6x^2+25x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+25x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.6.24 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.6} & & = \frac{-25+\sqrt49}{2.6} \\ & = \frac{-32}{12} & & = \frac{-18}{12} \\ & = \frac{-8}{3} & & = \frac{-3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{-3}{2} \Big\} & &\end{align} \\ -----------------\)
  8. \(x(x+46)=40(x+1) \\ \Leftrightarrow x^2+46x=40x+40 \\ \Leftrightarrow x^2+6x-40=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x-40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.(-40) & &\\ & = 36+160 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt196}{2.1} & & = \frac{-6+\sqrt196}{2.1} \\ & = \frac{-20}{2} & & = \frac{8}{2} \\ & = -10 & & = 4 \\ \\ V &= \Big\{ -10 ; 4 \Big\} & &\end{align} \\ -----------------\)
  9. \((2x+3)(-x-3)-x(-3x+1)=-73\\ \Leftrightarrow -2x^2-6x-3x-9 +3x^2-x+73=0 \\ \Leftrightarrow x^2-16x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.64 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.1} & & \\ & = 8 & & \\V &= \Big\{ 8 \Big\} & &\end{align} \\ -----------------\)
  10. \((4x+2)(4x-3)-x(15x-34)=-70\\ \Leftrightarrow 16x^2-12x+8x-6 -15x^2+34x+70=0 \\ \Leftrightarrow x^2+16x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.64 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.1} & & \\ & = -8 & & \\V &= \Big\{ -8 \Big\} & &\end{align} \\ -----------------\)
  11. \(\frac{1}{2}x^2-\frac{3}{4}x+\frac{81}{32}=0\\ \Leftrightarrow \color{red}{32.} \left(\frac{1}{2}x^2-\frac{3}{4}x+\frac{81}{32}\right)=0 \color{red}{.32} \\ \Leftrightarrow 16x^2-24x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-24x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.16.81 & &\\ & = 576-5184 & & \\ & = -4608 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  12. \(\frac{1}{10}x^2-\frac{1}{5}x-\frac{24}{5}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{1}{10}x^2-\frac{1}{5}x-\frac{24}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow x^2-2x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-48) & &\\ & = 4+192 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt196}{2.1} & & = \frac{-(-2)+\sqrt196}{2.1} \\ & = \frac{-12}{2} & & = \frac{16}{2} \\ & = -6 & & = 8 \\ \\ V &= \Big\{ -6 ; 8 \Big\} & &\end{align} \\ -----------------\)
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