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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \((-x-4)(-5x+1)-x(4x-11)=51\)
  2. \(\frac{2}{5}x^2+2x+\frac{5}{2}=0\)
  3. \(\frac{4}{45}x=-\frac{1}{5}x^2-\frac{16}{5}\)
  4. \(\frac{1}{8}x^2-x-\frac{5}{2}=0\)
  5. \(x(3x+1)=-12(x+1)\)
  6. \(17x^2-(6x-81)=x(x-78)\)
  7. \(17x^2-(7x-1)=x(x-24)\)
  8. \(\frac{25}{6}x=-12x^2-\frac{1}{3}\)
  9. \(-\frac{1}{3}x=-\frac{1}{3}x^2+\frac{110}{3}\)
  10. \(-\frac{7}{3}x=-\frac{1}{9}x^2-12\)
  11. \(x(x+89)=90(x+1)\)
  12. \(-(5-10x)=-x^2-(-39-17x)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \((-x-4)(-5x+1)-x(4x-11)=51\\ \Leftrightarrow 5x^2-x+20x-4 -4x^2+11x-51=0 \\ \Leftrightarrow x^2+6x-55=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x-55=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.(-55) & &\\ & = 36+220 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt256}{2.1} & & = \frac{-6+\sqrt256}{2.1} \\ & = \frac{-22}{2} & & = \frac{10}{2} \\ & = -11 & & = 5 \\ \\ V &= \Big\{ -11 ; 5 \Big\} & &\end{align} \\ -----------------\)
  2. \(\frac{2}{5}x^2+2x+\frac{5}{2}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{2}{5}x^2+2x+\frac{5}{2}\right)=0 \color{red}{.10} \\ \Leftrightarrow 4x^2+20x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+20x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.4.25 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.4} & & \\ & = -\frac{5}{2} & & \\V &= \Big\{ -\frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
  3. \(\frac{4}{45}x=-\frac{1}{5}x^2-\frac{16}{5} \\ \Leftrightarrow \frac{1}{5}x^2+\frac{4}{45}x+\frac{16}{5}=0 \\ \Leftrightarrow \color{red}{45.} \left(\frac{1}{5}x^2+\frac{4}{45}x+\frac{16}{5}\right)=0 \color{red}{.45} \\ \Leftrightarrow 9x^2+4x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+4x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.9.144 & &\\ & = 16-5184 & & \\ & = -5168 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  4. \(\frac{1}{8}x^2-x-\frac{5}{2}=0\\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2-x-\frac{5}{2}\right)=0 \color{red}{.8} \\ \Leftrightarrow x^2-8x-20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-20) & &\\ & = 64+80 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt144}{2.1} & & = \frac{-(-8)+\sqrt144}{2.1} \\ & = \frac{-4}{2} & & = \frac{20}{2} \\ & = -2 & & = 10 \\ \\ V &= \Big\{ -2 ; 10 \Big\} & &\end{align} \\ -----------------\)
  5. \(x(3x+1)=-12(x+1) \\ \Leftrightarrow 3x^2+x=-12x-12 \\ \Leftrightarrow 3x^2+13x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+13x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.3.12 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.3} & & = \frac{-13+\sqrt25}{2.3} \\ & = \frac{-18}{6} & & = \frac{-8}{6} \\ & = -3 & & = \frac{-4}{3} \\ \\ V &= \Big\{ -3 ; \frac{-4}{3} \Big\} & &\end{align} \\ -----------------\)
  6. \(17x^2-(6x-81)=x(x-78) \\ \Leftrightarrow 17x^2-6x+81=x^2-78x \\ \Leftrightarrow 16x^2+72x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+72x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (72)^2-4.16.81 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-72}{2.16} & & \\ & = -\frac{9}{4} & & \\V &= \Big\{ -\frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
  7. \(17x^2-(7x-1)=x(x-24) \\ \Leftrightarrow 17x^2-7x+1=x^2-24x \\ \Leftrightarrow 16x^2+17x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+17x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.16.1 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.16} & & = \frac{-17+\sqrt225}{2.16} \\ & = \frac{-32}{32} & & = \frac{-2}{32} \\ & = -1 & & = \frac{-1}{16} \\ \\ V &= \Big\{ -1 ; \frac{-1}{16} \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{25}{6}x=-12x^2-\frac{1}{3} \\ \Leftrightarrow 12x^2+\frac{25}{6}x+\frac{1}{3}=0 \\ \Leftrightarrow \color{red}{6.} \left(12x^2+\frac{25}{6}x+\frac{1}{3}\right)=0 \color{red}{.6} \\ \Leftrightarrow 72x^2+25x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+25x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.72.2 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.72} & & = \frac{-25+\sqrt49}{2.72} \\ & = \frac{-32}{144} & & = \frac{-18}{144} \\ & = \frac{-2}{9} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
  9. \(-\frac{1}{3}x=-\frac{1}{3}x^2+\frac{110}{3} \\ \Leftrightarrow \frac{1}{3}x^2-\frac{1}{3}x-\frac{110}{3}=0 \\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2-\frac{1}{3}x-\frac{110}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2-x-110=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-110=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-110) & &\\ & = 1+440 & & \\ & = 441 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt441}{2.1} & & = \frac{-(-1)+\sqrt441}{2.1} \\ & = \frac{-20}{2} & & = \frac{22}{2} \\ & = -10 & & = 11 \\ \\ V &= \Big\{ -10 ; 11 \Big\} & &\end{align} \\ -----------------\)
  10. \(-\frac{7}{3}x=-\frac{1}{9}x^2-12 \\ \Leftrightarrow \frac{1}{9}x^2-\frac{7}{3}x+12=0 \\ \Leftrightarrow \color{red}{9.} \left(\frac{1}{9}x^2-\frac{7}{3}x+12\right)=0 \color{red}{.9} \\ \Leftrightarrow x^2-21x+108=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-21x+108=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-21)^2-4.1.108 & &\\ & = 441-432 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-21)-\sqrt9}{2.1} & & = \frac{-(-21)+\sqrt9}{2.1} \\ & = \frac{18}{2} & & = \frac{24}{2} \\ & = 9 & & = 12 \\ \\ V &= \Big\{ 9 ; 12 \Big\} & &\end{align} \\ -----------------\)
  11. \(x(x+89)=90(x+1) \\ \Leftrightarrow x^2+89x=90x+90 \\ \Leftrightarrow x^2-x-90=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-90=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-90) & &\\ & = 1+360 & & \\ & = 361 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt361}{2.1} & & = \frac{-(-1)+\sqrt361}{2.1} \\ & = \frac{-18}{2} & & = \frac{20}{2} \\ & = -9 & & = 10 \\ \\ V &= \Big\{ -9 ; 10 \Big\} & &\end{align} \\ -----------------\)
  12. \(-(5-10x)=-x^2-(-39-17x) \\ \Leftrightarrow -5+10x=-x^2+39+17x \\ \Leftrightarrow x^2-7x-44=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x-44=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.(-44) & &\\ & = 49+176 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt225}{2.1} & & = \frac{-(-7)+\sqrt225}{2.1} \\ & = \frac{-8}{2} & & = \frac{22}{2} \\ & = -4 & & = 11 \\ \\ V &= \Big\{ -4 ; 11 \Big\} & &\end{align} \\ -----------------\)
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