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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{4}{3}x^2+\frac{7}{6}x-3=0\)
  2. \((4x+1)(5x-3)-x(19x-16)=87\)
  3. \(-(3+13x)=-x^2-(87-6x)\)
  4. \(\frac{1}{3}x^2-\frac{13}{3}x+10=0\)
  5. \(-(13-16x)=-x^2-(22-10x)\)
  6. \(x(16x+1)=5(x-20)\)
  7. \(\frac{25}{16}x=-\frac{9}{4}x^2-\frac{1}{4}\)
  8. \(2x=-\frac{3}{7}x^2-\frac{7}{3}\)
  9. \(x(4x-149)=-121(x+1)\)
  10. \((-3x+1)(4x-2)-x(-28x-11)=-1\)
  11. \(x(x+17)=2(x-27)\)
  12. \(10x^2-(8x-49)=x(x+34)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{4}{3}x^2+\frac{7}{6}x-3=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{4}{3}x^2+\frac{7}{6}x-3\right)=0 \color{red}{.6} \\ \Leftrightarrow 8x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.8.(-18) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\ & = \frac{-32}{16} & & = \frac{18}{16} \\ & = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)
  2. \((4x+1)(5x-3)-x(19x-16)=87\\ \Leftrightarrow 20x^2-12x+5x-3 -19x^2+16x-87=0 \\ \Leftrightarrow x^2+x-90=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-90=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-90) & &\\ & = 1+360 & & \\ & = 361 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt361}{2.1} & & = \frac{-1+\sqrt361}{2.1} \\ & = \frac{-20}{2} & & = \frac{18}{2} \\ & = -10 & & = 9 \\ \\ V &= \Big\{ -10 ; 9 \Big\} & &\end{align} \\ -----------------\)
  3. \(-(3+13x)=-x^2-(87-6x) \\ \Leftrightarrow -3-13x=-x^2-87+6x \\ \Leftrightarrow x^2-19x+84=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-19x+84=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-19)^2-4.1.84 & &\\ & = 361-336 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-19)-\sqrt25}{2.1} & & = \frac{-(-19)+\sqrt25}{2.1} \\ & = \frac{14}{2} & & = \frac{24}{2} \\ & = 7 & & = 12 \\ \\ V &= \Big\{ 7 ; 12 \Big\} & &\end{align} \\ -----------------\)
  4. \(\frac{1}{3}x^2-\frac{13}{3}x+10=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2-\frac{13}{3}x+10\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2-13x+30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-13x+30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-13)^2-4.1.30 & &\\ & = 169-120 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-13)-\sqrt49}{2.1} & & = \frac{-(-13)+\sqrt49}{2.1} \\ & = \frac{6}{2} & & = \frac{20}{2} \\ & = 3 & & = 10 \\ \\ V &= \Big\{ 3 ; 10 \Big\} & &\end{align} \\ -----------------\)
  5. \(-(13-16x)=-x^2-(22-10x) \\ \Leftrightarrow -13+16x=-x^2-22+10x \\ \Leftrightarrow x^2+6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.9 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-6}{2.1} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
  6. \(x(16x+1)=5(x-20) \\ \Leftrightarrow 16x^2+x=5x-100 \\ \Leftrightarrow 16x^2-4x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-4x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.16.100 & &\\ & = 16-6400 & & \\ & = -6384 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  7. \(\frac{25}{16}x=-\frac{9}{4}x^2-\frac{1}{4} \\ \Leftrightarrow \frac{9}{4}x^2+\frac{25}{16}x+\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{16.} \left(\frac{9}{4}x^2+\frac{25}{16}x+\frac{1}{4}\right)=0 \color{red}{.16} \\ \Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  8. \(2x=-\frac{3}{7}x^2-\frac{7}{3} \\ \Leftrightarrow \frac{3}{7}x^2+2x+\frac{7}{3}=0 \\ \Leftrightarrow \color{red}{21.} \left(\frac{3}{7}x^2+2x+\frac{7}{3}\right)=0 \color{red}{.21} \\ \Leftrightarrow 9x^2+42x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+42x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (42)^2-4.9.49 & &\\ & = 1764-1764 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-42}{2.9} & & \\ & = -\frac{7}{3} & & \\V &= \Big\{ -\frac{7}{3} \Big\} & &\end{align} \\ -----------------\)
  9. \(x(4x-149)=-121(x+1) \\ \Leftrightarrow 4x^2-149x=-121x-121 \\ \Leftrightarrow 4x^2-28x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-28x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-28)^2-4.4.121 & &\\ & = 784-1936 & & \\ & = -1152 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  10. \((-3x+1)(4x-2)-x(-28x-11)=-1\\ \Leftrightarrow -12x^2+6x+4x-2 +28x^2+11x+1=0 \\ \Leftrightarrow 16x^2+15x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+15x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.16.(-1) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.16} & & = \frac{-15+\sqrt289}{2.16} \\ & = \frac{-32}{32} & & = \frac{2}{32} \\ & = -1 & & = \frac{1}{16} \\ \\ V &= \Big\{ -1 ; \frac{1}{16} \Big\} & &\end{align} \\ -----------------\)
  11. \(x(x+17)=2(x-27) \\ \Leftrightarrow x^2+17x=2x-54 \\ \Leftrightarrow x^2+15x+54=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+15x+54=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.1.54 & &\\ & = 225-216 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt9}{2.1} & & = \frac{-15+\sqrt9}{2.1} \\ & = \frac{-18}{2} & & = \frac{-12}{2} \\ & = -9 & & = -6 \\ \\ V &= \Big\{ -9 ; -6 \Big\} & &\end{align} \\ -----------------\)
  12. \(10x^2-(8x-49)=x(x+34) \\ \Leftrightarrow 10x^2-8x+49=x^2+34x \\ \Leftrightarrow 9x^2-42x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-42x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-42)^2-4.9.49 & &\\ & = 1764-1764 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-42)}{2.9} & & \\ & = \frac{7}{3} & & \\V &= \Big\{ \frac{7}{3} \Big\} & &\end{align} \\ -----------------\)
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