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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(x(4x+19)=2(x-2)\)
  2. \(-\frac{1}{5}x=-\frac{1}{60}x^2-\frac{3}{5}\)
  3. \((-5x+3)(4x+1)-x(-21x-7)=9\)
  4. \(\frac{1}{2}x^2+\frac{13}{6}x+2=0\)
  5. \((-x+3)(4x-2)-x(-5x-2)=57\)
  6. \(x(3x+31)=6(x-8)\)
  7. \(x(16x+16)=(x+1)\)
  8. \(-\frac{3}{2}x=-\frac{9}{40}x^2-\frac{5}{2}\)
  9. \(\frac{1}{4}x^2+x-\frac{21}{4}=0\)
  10. \(x(x-15)=9(x-16)\)
  11. \((3x+2)(-2x+4)-x(-7x+34)=-41\)
  12. \((-5x-3)(-4x+5)-x(19x-35)=21\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(x(4x+19)=2(x-2) \\ \Leftrightarrow 4x^2+19x=2x-4 \\ \Leftrightarrow 4x^2+17x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+17x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.4.4 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.4} & & = \frac{-17+\sqrt225}{2.4} \\ & = \frac{-32}{8} & & = \frac{-2}{8} \\ & = -4 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -4 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  2. \(-\frac{1}{5}x=-\frac{1}{60}x^2-\frac{3}{5} \\ \Leftrightarrow \frac{1}{60}x^2-\frac{1}{5}x+\frac{3}{5}=0 \\ \Leftrightarrow \color{red}{60.} \left(\frac{1}{60}x^2-\frac{1}{5}x+\frac{3}{5}\right)=0 \color{red}{.60} \\ \Leftrightarrow x^2-12x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.36 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-12)}{2.1} & & \\ & = 6 & & \\V &= \Big\{ 6 \Big\} & &\end{align} \\ -----------------\)
  3. \((-5x+3)(4x+1)-x(-21x-7)=9\\ \Leftrightarrow -20x^2-5x+12x+3 +21x^2+7x-9=0 \\ \Leftrightarrow x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-6) & &\\ & = 25+24 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt49}{2.1} & & = \frac{-5+\sqrt49}{2.1} \\ & = \frac{-12}{2} & & = \frac{2}{2} \\ & = -6 & & = 1 \\ \\ V &= \Big\{ -6 ; 1 \Big\} & &\end{align} \\ -----------------\)
  4. \(\frac{1}{2}x^2+\frac{13}{6}x+2=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{2}x^2+\frac{13}{6}x+2\right)=0 \color{red}{.6} \\ \Leftrightarrow 3x^2+13x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+13x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.3.12 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.3} & & = \frac{-13+\sqrt25}{2.3} \\ & = \frac{-18}{6} & & = \frac{-8}{6} \\ & = -3 & & = \frac{-4}{3} \\ \\ V &= \Big\{ -3 ; \frac{-4}{3} \Big\} & &\end{align} \\ -----------------\)
  5. \((-x+3)(4x-2)-x(-5x-2)=57\\ \Leftrightarrow -4x^2+2x+12x-6 +5x^2+2x-57=0 \\ \Leftrightarrow x^2-2x-63=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-63=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-63) & &\\ & = 4+252 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt256}{2.1} & & = \frac{-(-2)+\sqrt256}{2.1} \\ & = \frac{-14}{2} & & = \frac{18}{2} \\ & = -7 & & = 9 \\ \\ V &= \Big\{ -7 ; 9 \Big\} & &\end{align} \\ -----------------\)
  6. \(x(3x+31)=6(x-8) \\ \Leftrightarrow 3x^2+31x=6x-48 \\ \Leftrightarrow 3x^2+25x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+25x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.3.48 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.3} & & = \frac{-25+\sqrt49}{2.3} \\ & = \frac{-32}{6} & & = \frac{-18}{6} \\ & = \frac{-16}{3} & & = -3 \\ \\ V &= \Big\{ \frac{-16}{3} ; -3 \Big\} & &\end{align} \\ -----------------\)
  7. \(x(16x+16)=(x+1) \\ \Leftrightarrow 16x^2+16x=x+1 \\ \Leftrightarrow 16x^2+15x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+15x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.16.(-1) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.16} & & = \frac{-15+\sqrt289}{2.16} \\ & = \frac{-32}{32} & & = \frac{2}{32} \\ & = -1 & & = \frac{1}{16} \\ \\ V &= \Big\{ -1 ; \frac{1}{16} \Big\} & &\end{align} \\ -----------------\)
  8. \(-\frac{3}{2}x=-\frac{9}{40}x^2-\frac{5}{2} \\ \Leftrightarrow \frac{9}{40}x^2-\frac{3}{2}x+\frac{5}{2}=0 \\ \Leftrightarrow \color{red}{40.} \left(\frac{9}{40}x^2-\frac{3}{2}x+\frac{5}{2}\right)=0 \color{red}{.40} \\ \Leftrightarrow 9x^2-60x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-60x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-60)^2-4.9.100 & &\\ & = 3600-3600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-60)}{2.9} & & \\ & = \frac{10}{3} & & \\V &= \Big\{ \frac{10}{3} \Big\} & &\end{align} \\ -----------------\)
  9. \(\frac{1}{4}x^2+x-\frac{21}{4}=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+x-\frac{21}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+4x-21=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-21=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-21) & &\\ & = 16+84 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt100}{2.1} & & = \frac{-4+\sqrt100}{2.1} \\ & = \frac{-14}{2} & & = \frac{6}{2} \\ & = -7 & & = 3 \\ \\ V &= \Big\{ -7 ; 3 \Big\} & &\end{align} \\ -----------------\)
  10. \(x(x-15)=9(x-16) \\ \Leftrightarrow x^2-15x=9x-144 \\ \Leftrightarrow x^2-24x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-24x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.1.144 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-24)}{2.1} & & \\ & = 12 & & \\V &= \Big\{ 12 \Big\} & &\end{align} \\ -----------------\)
  11. \((3x+2)(-2x+4)-x(-7x+34)=-41\\ \Leftrightarrow -6x^2+12x-4x+8 +7x^2-34x+41=0 \\ \Leftrightarrow x^2-14x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.49 & &\\ & = 196-196 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-14)}{2.1} & & \\ & = 7 & & \\V &= \Big\{ 7 \Big\} & &\end{align} \\ -----------------\)
  12. \((-5x-3)(-4x+5)-x(19x-35)=21\\ \Leftrightarrow 20x^2-25x+12x-15 -19x^2+35x-21=0 \\ \Leftrightarrow x^2-5x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-36) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt169}{2.1} & & = \frac{-(-5)+\sqrt169}{2.1} \\ & = \frac{-8}{2} & & = \frac{18}{2} \\ & = -4 & & = 9 \\ \\ V &= \Big\{ -4 ; 9 \Big\} & &\end{align} \\ -----------------\)
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