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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(x(9x+16)=8(x-2)\)
  2. \(-(11-9x)=-x^2-(19-3x)\)
  3. \(\frac{13}{9}x=-\frac{4}{3}x^2-\frac{1}{3}\)
  4. \(\frac{4}{5}x^2+x+\frac{1}{5}=0\)
  5. \((-3x+2)(-5x-2)-x(11x+42)=-104\)
  6. \(3x^2+\frac{7}{6}x-\frac{4}{3}=0\)
  7. \(\frac{1}{5}x^2+\frac{7}{10}x-\frac{36}{5}=0\)
  8. \(\frac{1}{2}x=-\frac{9}{5}x^2+\frac{1}{5}\)
  9. \(x(x-11)=-5(x+1)\)
  10. \((2x+5)(-5x+5)-x(-14x+18)=21\)
  11. \(x(x-11)=3(x-11)\)
  12. \(x(18x+11)=-2(x+1)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(x(9x+16)=8(x-2) \\ \Leftrightarrow 9x^2+16x=8x-16 \\ \Leftrightarrow 9x^2+8x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+8x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.9.16 & &\\ & = 64-576 & & \\ & = -512 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  2. \(-(11-9x)=-x^2-(19-3x) \\ \Leftrightarrow -11+9x=-x^2-19+3x \\ \Leftrightarrow x^2+6x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.8 & &\\ & = 36-32 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt4}{2.1} & & = \frac{-6+\sqrt4}{2.1} \\ & = \frac{-8}{2} & & = \frac{-4}{2} \\ & = -4 & & = -2 \\ \\ V &= \Big\{ -4 ; -2 \Big\} & &\end{align} \\ -----------------\)
  3. \(\frac{13}{9}x=-\frac{4}{3}x^2-\frac{1}{3} \\ \Leftrightarrow \frac{4}{3}x^2+\frac{13}{9}x+\frac{1}{3}=0 \\ \Leftrightarrow \color{red}{9.} \left(\frac{4}{3}x^2+\frac{13}{9}x+\frac{1}{3}\right)=0 \color{red}{.9} \\ \Leftrightarrow 12x^2+13x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+13x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.12.3 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.12} & & = \frac{-13+\sqrt25}{2.12} \\ & = \frac{-18}{24} & & = \frac{-8}{24} \\ & = \frac{-3}{4} & & = \frac{-1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{-1}{3} \Big\} & &\end{align} \\ -----------------\)
  4. \(\frac{4}{5}x^2+x+\frac{1}{5}=0\\ \Leftrightarrow \color{red}{5.} \left(\frac{4}{5}x^2+x+\frac{1}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow 4x^2+5x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.1 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.4} & & = \frac{-5+\sqrt9}{2.4} \\ & = \frac{-8}{8} & & = \frac{-2}{8} \\ & = -1 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -1 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  5. \((-3x+2)(-5x-2)-x(11x+42)=-104\\ \Leftrightarrow 15x^2+6x-10x-4 -11x^2-42x+104=0 \\ \Leftrightarrow 4x^2-40x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-40x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-40)^2-4.4.100 & &\\ & = 1600-1600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-40)}{2.4} & & \\ & = 5 & & \\V &= \Big\{ 5 \Big\} & &\end{align} \\ -----------------\)
  6. \(3x^2+\frac{7}{6}x-\frac{4}{3}=0\\ \Leftrightarrow \color{red}{6.} \left(3x^2+\frac{7}{6}x-\frac{4}{3}\right)=0 \color{red}{.6} \\ \Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  7. \(\frac{1}{5}x^2+\frac{7}{10}x-\frac{36}{5}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{1}{5}x^2+\frac{7}{10}x-\frac{36}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{1}{2}x=-\frac{9}{5}x^2+\frac{1}{5} \\ \Leftrightarrow \frac{9}{5}x^2+\frac{1}{2}x-\frac{1}{5}=0 \\ \Leftrightarrow \color{red}{10.} \left(\frac{9}{5}x^2+\frac{1}{2}x-\frac{1}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 18x^2+5x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+5x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.18.(-2) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.18} & & = \frac{-5+\sqrt169}{2.18} \\ & = \frac{-18}{36} & & = \frac{8}{36} \\ & = \frac{-1}{2} & & = \frac{2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{2}{9} \Big\} & &\end{align} \\ -----------------\)
  9. \(x(x-11)=-5(x+1) \\ \Leftrightarrow x^2-11x=-5x-5 \\ \Leftrightarrow x^2-6x+5=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+5=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.5 & &\\ & = 36-20 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt16}{2.1} & & = \frac{-(-6)+\sqrt16}{2.1} \\ & = \frac{2}{2} & & = \frac{10}{2} \\ & = 1 & & = 5 \\ \\ V &= \Big\{ 1 ; 5 \Big\} & &\end{align} \\ -----------------\)
  10. \((2x+5)(-5x+5)-x(-14x+18)=21\\ \Leftrightarrow -10x^2+10x-25x+25 +14x^2-18x-21=0 \\ \Leftrightarrow 4x^2+17x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+17x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.4.4 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.4} & & = \frac{-17+\sqrt225}{2.4} \\ & = \frac{-32}{8} & & = \frac{-2}{8} \\ & = -4 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -4 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  11. \(x(x-11)=3(x-11) \\ \Leftrightarrow x^2-11x=3x-33 \\ \Leftrightarrow x^2-14x+33=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+33=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.33 & &\\ & = 196-132 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-14)-\sqrt64}{2.1} & & = \frac{-(-14)+\sqrt64}{2.1} \\ & = \frac{6}{2} & & = \frac{22}{2} \\ & = 3 & & = 11 \\ \\ V &= \Big\{ 3 ; 11 \Big\} & &\end{align} \\ -----------------\)
  12. \(x(18x+11)=-2(x+1) \\ \Leftrightarrow 18x^2+11x=-2x-2 \\ \Leftrightarrow 18x^2+13x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+13x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.18.2 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.18} & & = \frac{-13+\sqrt25}{2.18} \\ & = \frac{-18}{36} & & = \frac{-8}{36} \\ & = \frac{-1}{2} & & = \frac{-2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-2}{9} \Big\} & &\end{align} \\ -----------------\)
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