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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(23x^2-(17x-1)=7x(x-3)\)
  2. \(16x^2-(16x+4)=7x(x-3)\)
  3. \(\frac{7}{4}x=-\frac{1}{2}x^2+18\)
  4. \(-(3-33x)=-16x^2-(7-17x)\)
  5. \(\frac{1}{3}x^2+\frac{7}{9}x-\frac{16}{3}=0\)
  6. \((x+5)(4x+2)-x(3x+18)=26\)
  7. \(x(9x+77)=11(x-11)\)
  8. \(\frac{1}{2}x=-\frac{1}{8}x^2-\frac{1}{2}\)
  9. \(-(15-4x)=-x^2-(-17-8x)\)
  10. \(x^2+\frac{10}{3}x+\frac{64}{9}=0\)
  11. \(21x^2-(7x-121)=5x(x-19)\)
  12. \(3x=-\frac{1}{5}x^2-\frac{44}{5}\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(23x^2-(17x-1)=7x(x-3) \\ \Leftrightarrow 23x^2-17x+1=7x^2-21x \\ \Leftrightarrow 16x^2+4x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+4x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.16.1 & &\\ & = 16-64 & & \\ & = -48 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  2. \(16x^2-(16x+4)=7x(x-3) \\ \Leftrightarrow 16x^2-16x-4=7x^2-21x \\ \Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.9.(-4) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\ & = \frac{-18}{18} & & = \frac{8}{18} \\ & = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)
  3. \(\frac{7}{4}x=-\frac{1}{2}x^2+18 \\ \Leftrightarrow \frac{1}{2}x^2+\frac{7}{4}x-18=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{2}x^2+\frac{7}{4}x-18\right)=0 \color{red}{.4} \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
  4. \(-(3-33x)=-16x^2-(7-17x) \\ \Leftrightarrow -3+33x=-16x^2-7+17x \\ \Leftrightarrow 16x^2+16x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+16x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.16.4 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.16} & & \\ & = -\frac{1}{2} & & \\V &= \Big\{ -\frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{1}{3}x^2+\frac{7}{9}x-\frac{16}{3}=0\\ \Leftrightarrow \color{red}{9.} \left(\frac{1}{3}x^2+\frac{7}{9}x-\frac{16}{3}\right)=0 \color{red}{.9} \\ \Leftrightarrow 3x^2+7x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+7x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.3.(-48) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.3} & & = \frac{-7+\sqrt625}{2.3} \\ & = \frac{-32}{6} & & = \frac{18}{6} \\ & = \frac{-16}{3} & & = 3 \\ \\ V &= \Big\{ \frac{-16}{3} ; 3 \Big\} & &\end{align} \\ -----------------\)
  6. \((x+5)(4x+2)-x(3x+18)=26\\ \Leftrightarrow 4x^2+2x+20x+10 -3x^2-18x-26=0 \\ \Leftrightarrow x^2-6x-16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.(-16) & &\\ & = 36+64 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt100}{2.1} & & = \frac{-(-6)+\sqrt100}{2.1} \\ & = \frac{-4}{2} & & = \frac{16}{2} \\ & = -2 & & = 8 \\ \\ V &= \Big\{ -2 ; 8 \Big\} & &\end{align} \\ -----------------\)
  7. \(x(9x+77)=11(x-11) \\ \Leftrightarrow 9x^2+77x=11x-121 \\ \Leftrightarrow 9x^2+66x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+66x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (66)^2-4.9.121 & &\\ & = 4356-4356 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-66}{2.9} & & \\ & = -\frac{11}{3} & & \\V &= \Big\{ -\frac{11}{3} \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{1}{2}x=-\frac{1}{8}x^2-\frac{1}{2} \\ \Leftrightarrow \frac{1}{8}x^2+\frac{1}{2}x+\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2+\frac{1}{2}x+\frac{1}{2}\right)=0 \color{red}{.8} \\ \Leftrightarrow x^2+4x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.1} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
  9. \(-(15-4x)=-x^2-(-17-8x) \\ \Leftrightarrow -15+4x=-x^2+17+8x \\ \Leftrightarrow x^2-4x-32=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-32=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-32) & &\\ & = 16+128 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt144}{2.1} & & = \frac{-(-4)+\sqrt144}{2.1} \\ & = \frac{-8}{2} & & = \frac{16}{2} \\ & = -4 & & = 8 \\ \\ V &= \Big\{ -4 ; 8 \Big\} & &\end{align} \\ -----------------\)
  10. \(x^2+\frac{10}{3}x+\frac{64}{9}=0\\ \Leftrightarrow \color{red}{9.} \left(x^2+\frac{10}{3}x+\frac{64}{9}\right)=0 \color{red}{.9} \\ \Leftrightarrow 9x^2+30x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+30x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (30)^2-4.9.64 & &\\ & = 900-2304 & & \\ & = -1404 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  11. \(21x^2-(7x-121)=5x(x-19) \\ \Leftrightarrow 21x^2-7x+121=5x^2-95x \\ \Leftrightarrow 16x^2+88x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+88x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (88)^2-4.16.121 & &\\ & = 7744-7744 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-88}{2.16} & & \\ & = -\frac{11}{4} & & \\V &= \Big\{ -\frac{11}{4} \Big\} & &\end{align} \\ -----------------\)
  12. \(3x=-\frac{1}{5}x^2-\frac{44}{5} \\ \Leftrightarrow \frac{1}{5}x^2+3x+\frac{44}{5}=0 \\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+3x+\frac{44}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2+15x+44=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+15x+44=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.1.44 & &\\ & = 225-176 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt49}{2.1} & & = \frac{-15+\sqrt49}{2.1} \\ & = \frac{-22}{2} & & = \frac{-8}{2} \\ & = -11 & & = -4 \\ \\ V &= \Big\{ -11 ; -4 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator vanhoeckes.be/wiskunde 2025-04-02 04:38:31