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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-(10-41x)=-18x^2-(18-16x)\)
  2. \(-\frac{7}{3}x=-\frac{1}{3}x^2+\frac{8}{3}\)
  3. \(x(x+1)=-(x+1)\)
  4. \(5x^2-(17x+12)=2x(x-11)\)
  5. \((x+5)(x+3)-x(-3x+13)=24\)
  6. \((-4x-3)(-4x+2)-x(12x-12)=-7\)
  7. \(39x^2-(14x+4)=3x(x-7)\)
  8. \(\frac{1}{3}x^2+\frac{7}{36}x-\frac{1}{3}=0\)
  9. \(-(3-34x)=-16x^2-(39-18x)\)
  10. \(17x^2-(8x-121)=x(x-20)\)
  11. \(10x^2-(13x-33)=9x(x-3)\)
  12. \(x(x+19)=24(x+1)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-(10-41x)=-18x^2-(18-16x) \\ \Leftrightarrow -10+41x=-18x^2-18+16x \\ \Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
  2. \(-\frac{7}{3}x=-\frac{1}{3}x^2+\frac{8}{3} \\ \Leftrightarrow \frac{1}{3}x^2-\frac{7}{3}x-\frac{8}{3}=0 \\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2-\frac{7}{3}x-\frac{8}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2-7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.(-8) & &\\ & = 49+32 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt81}{2.1} & & = \frac{-(-7)+\sqrt81}{2.1} \\ & = \frac{-2}{2} & & = \frac{16}{2} \\ & = -1 & & = 8 \\ \\ V &= \Big\{ -1 ; 8 \Big\} & &\end{align} \\ -----------------\)
  3. \(x(x+1)=-(x+1) \\ \Leftrightarrow x^2+x=-x-1 \\ \Leftrightarrow x^2+2x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.1 & &\\ & = 4-4 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-2}{2.1} & & \\ & = -1 & & \\V &= \Big\{ -1 \Big\} & &\end{align} \\ -----------------\)
  4. \(5x^2-(17x+12)=2x(x-11) \\ \Leftrightarrow 5x^2-17x-12=2x^2-22x \\ \Leftrightarrow 3x^2+5x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+5x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.3.(-12) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.3} & & = \frac{-5+\sqrt169}{2.3} \\ & = \frac{-18}{6} & & = \frac{8}{6} \\ & = -3 & & = \frac{4}{3} \\ \\ V &= \Big\{ -3 ; \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
  5. \((x+5)(x+3)-x(-3x+13)=24\\ \Leftrightarrow x^2+3x+5x+15 +3x^2-13x-24=0 \\ \Leftrightarrow 4x^2+5x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.(-9) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.4} & & = \frac{-5+\sqrt169}{2.4} \\ & = \frac{-18}{8} & & = \frac{8}{8} \\ & = \frac{-9}{4} & & = 1 \\ \\ V &= \Big\{ \frac{-9}{4} ; 1 \Big\} & &\end{align} \\ -----------------\)
  6. \((-4x-3)(-4x+2)-x(12x-12)=-7\\ \Leftrightarrow 16x^2-8x+12x-6 -12x^2+12x+7=0 \\ \Leftrightarrow 4x^2-2x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-2x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.4.1 & &\\ & = 4-16 & & \\ & = -12 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  7. \(39x^2-(14x+4)=3x(x-7) \\ \Leftrightarrow 39x^2-14x-4=3x^2-21x \\ \Leftrightarrow 36x^2+7x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+7x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.36.(-4) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.36} & & = \frac{-7+\sqrt625}{2.36} \\ & = \frac{-32}{72} & & = \frac{18}{72} \\ & = \frac{-4}{9} & & = \frac{1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{1}{3}x^2+\frac{7}{36}x-\frac{1}{3}=0\\ \Leftrightarrow \color{red}{36.} \left(\frac{1}{3}x^2+\frac{7}{36}x-\frac{1}{3}\right)=0 \color{red}{.36} \\ \Leftrightarrow 12x^2+7x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+7x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.12.(-12) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.12} & & = \frac{-7+\sqrt625}{2.12} \\ & = \frac{-32}{24} & & = \frac{18}{24} \\ & = \frac{-4}{3} & & = \frac{3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
  9. \(-(3-34x)=-16x^2-(39-18x) \\ \Leftrightarrow -3+34x=-16x^2-39+18x \\ \Leftrightarrow 16x^2+16x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+16x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.16.36 & &\\ & = 256-2304 & & \\ & = -2048 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  10. \(17x^2-(8x-121)=x(x-20) \\ \Leftrightarrow 17x^2-8x+121=x^2-20x \\ \Leftrightarrow 16x^2+12x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+12x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.16.121 & &\\ & = 144-7744 & & \\ & = -7600 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  11. \(10x^2-(13x-33)=9x(x-3) \\ \Leftrightarrow 10x^2-13x+33=9x^2-27x \\ \Leftrightarrow x^2+14x+33=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+33=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.33 & &\\ & = 196-132 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-14-\sqrt64}{2.1} & & = \frac{-14+\sqrt64}{2.1} \\ & = \frac{-22}{2} & & = \frac{-6}{2} \\ & = -11 & & = -3 \\ \\ V &= \Big\{ -11 ; -3 \Big\} & &\end{align} \\ -----------------\)
  12. \(x(x+19)=24(x+1) \\ \Leftrightarrow x^2+19x=24x+24 \\ \Leftrightarrow x^2-5x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-24) & &\\ & = 25+96 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt121}{2.1} & & = \frac{-(-5)+\sqrt121}{2.1} \\ & = \frac{-6}{2} & & = \frac{16}{2} \\ & = -3 & & = 8 \\ \\ V &= \Big\{ -3 ; 8 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator vanhoeckes.be/wiskunde 2024-05-08 04:36:44