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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{1}{5}x^2+\frac{17}{20}x+\frac{1}{5}=0\)
  2. \(2x^2-(11x+84)=x(x-6)\)
  3. \(\frac{3}{4}x=-\frac{1}{4}x^2-\frac{9}{16}\)
  4. \((-x+1)(5x-5)-x(-7x-15)=3\)
  5. \(\frac{1}{4}x^2+4x+\frac{63}{4}=0\)
  6. \(3x^2+\frac{13}{6}x+\frac{1}{3}=0\)
  7. \(-(6-40x)=-4x^2-(55-12x)\)
  8. \(-(2-13x)=-3x^2-(-46-6x)\)
  9. \(2x^2-(9x+24)=x(x-4)\)
  10. \(x(9x+9)=4(x+1)\)
  11. \(x(18x+15)=8(x+1)\)
  12. \(16x^2-(19x-3)=4x(x-8)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{1}{5}x^2+\frac{17}{20}x+\frac{1}{5}=0\\ \Leftrightarrow \color{red}{20.} \left(\frac{1}{5}x^2+\frac{17}{20}x+\frac{1}{5}\right)=0 \color{red}{.20} \\ \Leftrightarrow 4x^2+17x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+17x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.4.4 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.4} & & = \frac{-17+\sqrt225}{2.4} \\ & = \frac{-32}{8} & & = \frac{-2}{8} \\ & = -4 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -4 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  2. \(2x^2-(11x+84)=x(x-6) \\ \Leftrightarrow 2x^2-11x-84=x^2-6x \\ \Leftrightarrow x^2-5x-84=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-84=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-84) & &\\ & = 25+336 & & \\ & = 361 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt361}{2.1} & & = \frac{-(-5)+\sqrt361}{2.1} \\ & = \frac{-14}{2} & & = \frac{24}{2} \\ & = -7 & & = 12 \\ \\ V &= \Big\{ -7 ; 12 \Big\} & &\end{align} \\ -----------------\)
  3. \(\frac{3}{4}x=-\frac{1}{4}x^2-\frac{9}{16} \\ \Leftrightarrow \frac{1}{4}x^2+\frac{3}{4}x+\frac{9}{16}=0 \\ \Leftrightarrow \color{red}{16.} \left(\frac{1}{4}x^2+\frac{3}{4}x+\frac{9}{16}\right)=0 \color{red}{.16} \\ \Leftrightarrow 4x^2+12x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+12x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.4.9 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-12}{2.4} & & \\ & = -\frac{3}{2} & & \\V &= \Big\{ -\frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
  4. \((-x+1)(5x-5)-x(-7x-15)=3\\ \Leftrightarrow -5x^2+5x+5x-5 +7x^2+15x-3=0 \\ \Leftrightarrow 2x^2+15x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+15x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.2.(-8) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.2} & & = \frac{-15+\sqrt289}{2.2} \\ & = \frac{-32}{4} & & = \frac{2}{4} \\ & = -8 & & = \frac{1}{2} \\ \\ V &= \Big\{ -8 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{1}{4}x^2+4x+\frac{63}{4}=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+4x+\frac{63}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+16x+63=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+63=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.63 & &\\ & = 256-252 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-16-\sqrt4}{2.1} & & = \frac{-16+\sqrt4}{2.1} \\ & = \frac{-18}{2} & & = \frac{-14}{2} \\ & = -9 & & = -7 \\ \\ V &= \Big\{ -9 ; -7 \Big\} & &\end{align} \\ -----------------\)
  6. \(3x^2+\frac{13}{6}x+\frac{1}{3}=0\\ \Leftrightarrow \color{red}{6.} \left(3x^2+\frac{13}{6}x+\frac{1}{3}\right)=0 \color{red}{.6} \\ \Leftrightarrow 18x^2+13x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+13x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.18.2 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.18} & & = \frac{-13+\sqrt25}{2.18} \\ & = \frac{-18}{36} & & = \frac{-8}{36} \\ & = \frac{-1}{2} & & = \frac{-2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-2}{9} \Big\} & &\end{align} \\ -----------------\)
  7. \(-(6-40x)=-4x^2-(55-12x) \\ \Leftrightarrow -6+40x=-4x^2-55+12x \\ \Leftrightarrow 4x^2+28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+28x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (28)^2-4.4.49 & &\\ & = 784-784 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-28}{2.4} & & \\ & = -\frac{7}{2} & & \\V &= \Big\{ -\frac{7}{2} \Big\} & &\end{align} \\ -----------------\)
  8. \(-(2-13x)=-3x^2-(-46-6x) \\ \Leftrightarrow -2+13x=-3x^2+46+6x \\ \Leftrightarrow 3x^2+7x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+7x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.3.(-48) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.3} & & = \frac{-7+\sqrt625}{2.3} \\ & = \frac{-32}{6} & & = \frac{18}{6} \\ & = \frac{-16}{3} & & = 3 \\ \\ V &= \Big\{ \frac{-16}{3} ; 3 \Big\} & &\end{align} \\ -----------------\)
  9. \(2x^2-(9x+24)=x(x-4) \\ \Leftrightarrow 2x^2-9x-24=x^2-4x \\ \Leftrightarrow x^2-5x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-24) & &\\ & = 25+96 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt121}{2.1} & & = \frac{-(-5)+\sqrt121}{2.1} \\ & = \frac{-6}{2} & & = \frac{16}{2} \\ & = -3 & & = 8 \\ \\ V &= \Big\{ -3 ; 8 \Big\} & &\end{align} \\ -----------------\)
  10. \(x(9x+9)=4(x+1) \\ \Leftrightarrow 9x^2+9x=4x+4 \\ \Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.9.(-4) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\ & = \frac{-18}{18} & & = \frac{8}{18} \\ & = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)
  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. \(16x^2-(19x-3)=4x(x-8) \\ \Leftrightarrow 16x^2-19x+3=4x^2-32x \\ \Leftrightarrow 12x^2+13x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+13x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.12.3 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.12} & & = \frac{-13+\sqrt25}{2.12} \\ & = \frac{-18}{24} & & = \frac{-8}{24} \\ & = \frac{-3}{4} & & = \frac{-1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{-1}{3} \Big\} & &\end{align} \\ -----------------\)
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