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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{7}{15}x=-\frac{16}{5}x^2+\frac{1}{5}\)
  2. \(2x^2+\frac{25}{4}x+\frac{9}{2}=0\)
  3. \((x+4)(3x-4)-x(x-45)=-88\)
  4. \(-(14-30x)=-18x^2-(22-5x)\)
  5. \(-(6-28x)=-x^2-(22-18x)\)
  6. \(4x=-\frac{1}{3}x^2-\frac{35}{3}\)
  7. \(\frac{2}{5}x=-\frac{3}{5}x^2-\frac{1}{15}\)
  8. \((3x+4)(-x-2)-x(-7x-39)=-44\)
  9. \(3x^2-\frac{34}{3}x+\frac{121}{3}=0\)
  10. \(x(9x-55)=-121(x+1)\)
  11. \(\frac{1}{16}x^2-\frac{1}{4}x+\frac{1}{4}=0\)
  12. \((-5x-2)(x-3)-x(-21x+6)=7\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{7}{15}x=-\frac{16}{5}x^2+\frac{1}{5} \\ \Leftrightarrow \frac{16}{5}x^2+\frac{7}{15}x-\frac{1}{5}=0 \\ \Leftrightarrow \color{red}{15.} \left(\frac{16}{5}x^2+\frac{7}{15}x-\frac{1}{5}\right)=0 \color{red}{.15} \\ \Leftrightarrow 48x^2+7x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
  2. \(2x^2+\frac{25}{4}x+\frac{9}{2}=0\\ \Leftrightarrow \color{red}{4.} \left(2x^2+\frac{25}{4}x+\frac{9}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow 8x^2+25x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+25x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.8.18 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.8} & & = \frac{-25+\sqrt49}{2.8} \\ & = \frac{-32}{16} & & = \frac{-18}{16} \\ & = -2 & & = \frac{-9}{8} \\ \\ V &= \Big\{ -2 ; \frac{-9}{8} \Big\} & &\end{align} \\ -----------------\)
  3. \((x+4)(3x-4)-x(x-45)=-88\\ \Leftrightarrow 3x^2-4x+12x-16 -x^2+45x+88=0 \\ \Leftrightarrow 2x^2+25x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+25x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.2.72 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.2} & & = \frac{-25+\sqrt49}{2.2} \\ & = \frac{-32}{4} & & = \frac{-18}{4} \\ & = -8 & & = \frac{-9}{2} \\ \\ V &= \Big\{ -8 ; \frac{-9}{2} \Big\} & &\end{align} \\ -----------------\)
  4. \(-(14-30x)=-18x^2-(22-5x) \\ \Leftrightarrow -14+30x=-18x^2-22+5x \\ \Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
  5. \(-(6-28x)=-x^2-(22-18x) \\ \Leftrightarrow -6+28x=-x^2-22+18x \\ \Leftrightarrow x^2+10x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+10x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.1.16 & &\\ & = 100-64 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt36}{2.1} & & = \frac{-10+\sqrt36}{2.1} \\ & = \frac{-16}{2} & & = \frac{-4}{2} \\ & = -8 & & = -2 \\ \\ V &= \Big\{ -8 ; -2 \Big\} & &\end{align} \\ -----------------\)
  6. \(4x=-\frac{1}{3}x^2-\frac{35}{3} \\ \Leftrightarrow \frac{1}{3}x^2+4x+\frac{35}{3}=0 \\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+4x+\frac{35}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2+12x+35=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+35=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.35 & &\\ & = 144-140 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-12-\sqrt4}{2.1} & & = \frac{-12+\sqrt4}{2.1} \\ & = \frac{-14}{2} & & = \frac{-10}{2} \\ & = -7 & & = -5 \\ \\ V &= \Big\{ -7 ; -5 \Big\} & &\end{align} \\ -----------------\)
  7. \(\frac{2}{5}x=-\frac{3}{5}x^2-\frac{1}{15} \\ \Leftrightarrow \frac{3}{5}x^2+\frac{2}{5}x+\frac{1}{15}=0 \\ \Leftrightarrow \color{red}{15.} \left(\frac{3}{5}x^2+\frac{2}{5}x+\frac{1}{15}\right)=0 \color{red}{.15} \\ \Leftrightarrow 9x^2+6x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+6x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.9.1 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-6}{2.9} & & \\ & = -\frac{1}{3} & & \\V &= \Big\{ -\frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
  8. \((3x+4)(-x-2)-x(-7x-39)=-44\\ \Leftrightarrow -3x^2-6x-4x-8 +7x^2+39x+44=0 \\ \Leftrightarrow 4x^2+25x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+25x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.4.36 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.4} & & = \frac{-25+\sqrt49}{2.4} \\ & = \frac{-32}{8} & & = \frac{-18}{8} \\ & = -4 & & = \frac{-9}{4} \\ \\ V &= \Big\{ -4 ; \frac{-9}{4} \Big\} & &\end{align} \\ -----------------\)
  9. \(3x^2-\frac{34}{3}x+\frac{121}{3}=0\\ \Leftrightarrow \color{red}{3.} \left(3x^2-\frac{34}{3}x+\frac{121}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow 9x^2-34x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-34x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-34)^2-4.9.121 & &\\ & = 1156-4356 & & \\ & = -3200 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  10. \(x(9x-55)=-121(x+1) \\ \Leftrightarrow 9x^2-55x=-121x-121 \\ \Leftrightarrow 9x^2+66x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+66x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (66)^2-4.9.121 & &\\ & = 4356-4356 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-66}{2.9} & & \\ & = -\frac{11}{3} & & \\V &= \Big\{ -\frac{11}{3} \Big\} & &\end{align} \\ -----------------\)
  11. \(\frac{1}{16}x^2-\frac{1}{4}x+\frac{1}{4}=0\\ \Leftrightarrow \color{red}{16.} \left(\frac{1}{16}x^2-\frac{1}{4}x+\frac{1}{4}\right)=0 \color{red}{.16} \\ \Leftrightarrow x^2-4x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-4)}{2.1} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
  12. \((-5x-2)(x-3)-x(-21x+6)=7\\ \Leftrightarrow -5x^2+15x-2x+6 +21x^2-6x-7=0 \\ \Leftrightarrow 16x^2+15x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+15x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.16.(-1) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.16} & & = \frac{-15+\sqrt289}{2.16} \\ & = \frac{-32}{32} & & = \frac{2}{32} \\ & = -1 & & = \frac{1}{16} \\ \\ V &= \Big\{ -1 ; \frac{1}{16} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2024-06-28 22:52:35
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