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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{25}{8}x=-\frac{1}{4}x^2-9\)
  2. \(-(14-17x)=-x^2-(9-13x)\)
  3. \(\frac{1}{4}x^2-\frac{13}{4}x+\frac{21}{2}=0\)
  4. \(-(2-15x)=-4x^2-(-34-8x)\)
  5. \(-\frac{8}{5}x=-\frac{1}{5}x^2+\frac{33}{5}\)
  6. \(2x^2-(5x-16)=x(x+3)\)
  7. \(-\frac{7}{2}x=-x^2-\frac{49}{16}\)
  8. \(-(8-16x)=-12x^2-(11-3x)\)
  9. \(x(72x+9)=2(x+1)\)
  10. \((-3x-5)(x-2)-x(-7x+20)=9\)
  11. \(x(x-12)=4(x-16)\)
  12. \(6x^2-(6x-121)=2x(x-5)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{25}{8}x=-\frac{1}{4}x^2-9 \\ \Leftrightarrow \frac{1}{4}x^2+\frac{25}{8}x+9=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{4}x^2+\frac{25}{8}x+9\right)=0 \color{red}{.8} \\ \Leftrightarrow 2x^2+25x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+25x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.2.72 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.2} & & = \frac{-25+\sqrt49}{2.2} \\ & = \frac{-32}{4} & & = \frac{-18}{4} \\ & = -8 & & = \frac{-9}{2} \\ \\ V &= \Big\{ -8 ; \frac{-9}{2} \Big\} & &\end{align} \\ -----------------\)
  2. \(-(14-17x)=-x^2-(9-13x) \\ \Leftrightarrow -14+17x=-x^2-9+13x \\ \Leftrightarrow x^2+4x-5=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-5=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-5) & &\\ & = 16+20 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt36}{2.1} & & = \frac{-4+\sqrt36}{2.1} \\ & = \frac{-10}{2} & & = \frac{2}{2} \\ & = -5 & & = 1 \\ \\ V &= \Big\{ -5 ; 1 \Big\} & &\end{align} \\ -----------------\)
  3. \(\frac{1}{4}x^2-\frac{13}{4}x+\frac{21}{2}=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{13}{4}x+\frac{21}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-13x+42=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-13x+42=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-13)^2-4.1.42 & &\\ & = 169-168 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-13)-\sqrt1}{2.1} & & = \frac{-(-13)+\sqrt1}{2.1} \\ & = \frac{12}{2} & & = \frac{14}{2} \\ & = 6 & & = 7 \\ \\ V &= \Big\{ 6 ; 7 \Big\} & &\end{align} \\ -----------------\)
  4. \(-(2-15x)=-4x^2-(-34-8x) \\ \Leftrightarrow -2+15x=-4x^2+34+8x \\ \Leftrightarrow 4x^2+7x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+7x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.4.(-36) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.4} & & = \frac{-7+\sqrt625}{2.4} \\ & = \frac{-32}{8} & & = \frac{18}{8} \\ & = -4 & & = \frac{9}{4} \\ \\ V &= \Big\{ -4 ; \frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
  5. \(-\frac{8}{5}x=-\frac{1}{5}x^2+\frac{33}{5} \\ \Leftrightarrow \frac{1}{5}x^2-\frac{8}{5}x-\frac{33}{5}=0 \\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2-\frac{8}{5}x-\frac{33}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2-8x-33=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-33=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-33) & &\\ & = 64+132 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt196}{2.1} & & = \frac{-(-8)+\sqrt196}{2.1} \\ & = \frac{-6}{2} & & = \frac{22}{2} \\ & = -3 & & = 11 \\ \\ V &= \Big\{ -3 ; 11 \Big\} & &\end{align} \\ -----------------\)
  6. \(2x^2-(5x-16)=x(x+3) \\ \Leftrightarrow 2x^2-5x+16=x^2+3x \\ \Leftrightarrow x^2-8x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.16 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-8)}{2.1} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
  7. \(-\frac{7}{2}x=-x^2-\frac{49}{16} \\ \Leftrightarrow x^2-\frac{7}{2}x+\frac{49}{16}=0 \\ \Leftrightarrow \color{red}{16.} \left(x^2-\frac{7}{2}x+\frac{49}{16}\right)=0 \color{red}{.16} \\ \Leftrightarrow 16x^2-56x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-56x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-56)^2-4.16.49 & &\\ & = 3136-3136 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-56)}{2.16} & & \\ & = \frac{7}{4} & & \\V &= \Big\{ \frac{7}{4} \Big\} & &\end{align} \\ -----------------\)
  8. \(-(8-16x)=-12x^2-(11-3x) \\ \Leftrightarrow -8+16x=-12x^2-11+3x \\ \Leftrightarrow 12x^2+13x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+13x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.12.3 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.12} & & = \frac{-13+\sqrt25}{2.12} \\ & = \frac{-18}{24} & & = \frac{-8}{24} \\ & = \frac{-3}{4} & & = \frac{-1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{-1}{3} \Big\} & &\end{align} \\ -----------------\)
  9. \(x(72x+9)=2(x+1) \\ \Leftrightarrow 72x^2+9x=2x+2 \\ \Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.72.(-2) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\ & = \frac{-32}{144} & & = \frac{18}{144} \\ & = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  10. \((-3x-5)(x-2)-x(-7x+20)=9\\ \Leftrightarrow -3x^2+6x-5x+10 +7x^2-20x-9=0 \\ \Leftrightarrow 4x^2-4x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-4x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.4.1 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-4)}{2.4} & & \\ & = \frac{1}{2} & & \\V &= \Big\{ \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  11. \(x(x-12)=4(x-16) \\ \Leftrightarrow x^2-12x=4x-64 \\ \Leftrightarrow x^2-16x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.64 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.1} & & \\ & = 8 & & \\V &= \Big\{ 8 \Big\} & &\end{align} \\ -----------------\)
  12. \(6x^2-(6x-121)=2x(x-5) \\ \Leftrightarrow 6x^2-6x+121=2x^2-10x \\ \Leftrightarrow 4x^2+4x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+4x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.4.121 & &\\ & = 16-1936 & & \\ & = -1920 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
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