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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \((-2x+1)(5x+2)-x(-18x-9)=20\)
  2. \(-\frac{23}{8}x=-\frac{1}{4}x^2-9\)
  3. \(\frac{25}{8}x=-\frac{9}{2}x^2-\frac{1}{2}\)
  4. \(-x=-\frac{1}{18}x^2-\frac{9}{2}\)
  5. \(\frac{5}{4}x=-2x^2-\frac{1}{8}\)
  6. \(\frac{2}{5}x^2+2x+\frac{5}{2}=0\)
  7. \((-5x-1)(-2x+2)-x(6x-25)=-11\)
  8. \(\frac{16}{3}x^2+\frac{7}{9}x-\frac{1}{3}=0\)
  9. \(x(3x+29)=4(x-12)\)
  10. \(2x^2-(13x+77)=x(x-17)\)
  11. \(x(18x+15)=8(x+1)\)
  12. \(\frac{3}{8}x^2-x+\frac{2}{3}=0\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \((-2x+1)(5x+2)-x(-18x-9)=20\\ \Leftrightarrow -10x^2-4x+5x+2 +18x^2+9x-20=0 \\ \Leftrightarrow 8x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.8.(-18) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\ & = \frac{-32}{16} & & = \frac{18}{16} \\ & = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)
  2. \(-\frac{23}{8}x=-\frac{1}{4}x^2-9 \\ \Leftrightarrow \frac{1}{4}x^2-\frac{23}{8}x+9=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{4}x^2-\frac{23}{8}x+9\right)=0 \color{red}{.8} \\ \Leftrightarrow 4x^2-46x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-46x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-46)^2-4.4.144 & &\\ & = 2116-2304 & & \\ & = -188 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  3. \(\frac{25}{8}x=-\frac{9}{2}x^2-\frac{1}{2} \\ \Leftrightarrow \frac{9}{2}x^2+\frac{25}{8}x+\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{9}{2}x^2+\frac{25}{8}x+\frac{1}{2}\right)=0 \color{red}{.8} \\ \Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  4. \(-x=-\frac{1}{18}x^2-\frac{9}{2} \\ \Leftrightarrow \frac{1}{18}x^2-x+\frac{9}{2}=0 \\ \Leftrightarrow \color{red}{18.} \left(\frac{1}{18}x^2-x+\frac{9}{2}\right)=0 \color{red}{.18} \\ \Leftrightarrow x^2-18x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-18x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-18)^2-4.1.81 & &\\ & = 324-324 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-18)}{2.1} & & \\ & = 9 & & \\V &= \Big\{ 9 \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{5}{4}x=-2x^2-\frac{1}{8} \\ \Leftrightarrow 2x^2+\frac{5}{4}x+\frac{1}{8}=0 \\ \Leftrightarrow \color{red}{8.} \left(2x^2+\frac{5}{4}x+\frac{1}{8}\right)=0 \color{red}{.8} \\ \Leftrightarrow 16x^2+10x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+10x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.16.1 & &\\ & = 100-64 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt36}{2.16} & & = \frac{-10+\sqrt36}{2.16} \\ & = \frac{-16}{32} & & = \frac{-4}{32} \\ & = \frac{-1}{2} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
  6. \(\frac{2}{5}x^2+2x+\frac{5}{2}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{2}{5}x^2+2x+\frac{5}{2}\right)=0 \color{red}{.10} \\ \Leftrightarrow 4x^2+20x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+20x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.4.25 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.4} & & \\ & = -\frac{5}{2} & & \\V &= \Big\{ -\frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
  7. \((-5x-1)(-2x+2)-x(6x-25)=-11\\ \Leftrightarrow 10x^2-10x+2x-2 -6x^2+25x+11=0 \\ \Leftrightarrow 4x^2+13x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+13x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.4.9 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.4} & & = \frac{-13+\sqrt25}{2.4} \\ & = \frac{-18}{8} & & = \frac{-8}{8} \\ & = \frac{-9}{4} & & = -1 \\ \\ V &= \Big\{ \frac{-9}{4} ; -1 \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{16}{3}x^2+\frac{7}{9}x-\frac{1}{3}=0\\ \Leftrightarrow \color{red}{9.} \left(\frac{16}{3}x^2+\frac{7}{9}x-\frac{1}{3}\right)=0 \color{red}{.9} \\ \Leftrightarrow 48x^2+7x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
  9. \(x(3x+29)=4(x-12) \\ \Leftrightarrow 3x^2+29x=4x-48 \\ \Leftrightarrow 3x^2+25x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+25x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.3.48 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.3} & & = \frac{-25+\sqrt49}{2.3} \\ & = \frac{-32}{6} & & = \frac{-18}{6} \\ & = \frac{-16}{3} & & = -3 \\ \\ V &= \Big\{ \frac{-16}{3} ; -3 \Big\} & &\end{align} \\ -----------------\)
  10. \(2x^2-(13x+77)=x(x-17) \\ \Leftrightarrow 2x^2-13x-77=x^2-17x \\ \Leftrightarrow x^2+4x-77=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-77=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-77) & &\\ & = 16+308 & & \\ & = 324 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt324}{2.1} & & = \frac{-4+\sqrt324}{2.1} \\ & = \frac{-22}{2} & & = \frac{14}{2} \\ & = -11 & & = 7 \\ \\ V &= \Big\{ -11 ; 7 \Big\} & &\end{align} \\ -----------------\)
  11. \(x(18x+15)=8(x+1) \\ \Leftrightarrow 18x^2+15x=8x+8 \\ \Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  12. \(\frac{3}{8}x^2-x+\frac{2}{3}=0\\ \Leftrightarrow \color{red}{24.} \left(\frac{3}{8}x^2-x+\frac{2}{3}\right)=0 \color{red}{.24} \\ \Leftrightarrow 9x^2-24x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-24x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.9.16 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-24)}{2.9} & & \\ & = \frac{4}{3} & & \\V &= \Big\{ \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
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