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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-(13-51x)=-4x^2-(157-3x)\)
  2. \(x(x-6)=10(x-6)\)
  3. \(-(4+2x)=-x^2-(92-17x)\)
  4. \(10x^2-(15x-25)=x(x+11)\)
  5. \(x(2x+5)=2(x+1)\)
  6. \(-(11+7x)=-4x^2-(47-5x)\)
  7. \((2x-1)(3x-4)-x(-12x-29)=-4\)
  8. \(x(16x-73)=-49(x+1)\)
  9. \(\frac{1}{4}x^2+\frac{13}{24}x+\frac{1}{4}=0\)
  10. \(x^2+\frac{5}{2}x+\frac{25}{16}=0\)
  11. \(\frac{1}{2}x^2+\frac{17}{2}x+35=0\)
  12. \((-x-3)(3x-3)-x(-6x+5)=57\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-(13-51x)=-4x^2-(157-3x) \\ \Leftrightarrow -13+51x=-4x^2-157+3x \\ \Leftrightarrow 4x^2+48x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+48x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (48)^2-4.4.144 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-48}{2.4} & & \\ & = -6 & & \\V &= \Big\{ -6 \Big\} & &\end{align} \\ -----------------\)
  2. \(x(x-6)=10(x-6) \\ \Leftrightarrow x^2-6x=10x-60 \\ \Leftrightarrow x^2-16x+60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.60 & &\\ & = 256-240 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-16)-\sqrt16}{2.1} & & = \frac{-(-16)+\sqrt16}{2.1} \\ & = \frac{12}{2} & & = \frac{20}{2} \\ & = 6 & & = 10 \\ \\ V &= \Big\{ 6 ; 10 \Big\} & &\end{align} \\ -----------------\)
  3. \(-(4+2x)=-x^2-(92-17x) \\ \Leftrightarrow -4-2x=-x^2-92+17x \\ \Leftrightarrow x^2-19x+88=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-19x+88=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-19)^2-4.1.88 & &\\ & = 361-352 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-19)-\sqrt9}{2.1} & & = \frac{-(-19)+\sqrt9}{2.1} \\ & = \frac{16}{2} & & = \frac{22}{2} \\ & = 8 & & = 11 \\ \\ V &= \Big\{ 8 ; 11 \Big\} & &\end{align} \\ -----------------\)
  4. \(10x^2-(15x-25)=x(x+11) \\ \Leftrightarrow 10x^2-15x+25=x^2+11x \\ \Leftrightarrow 9x^2-26x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-26x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-26)^2-4.9.25 & &\\ & = 676-900 & & \\ & = -224 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  5. \(x(2x+5)=2(x+1) \\ \Leftrightarrow 2x^2+5x=2x+2 \\ \Leftrightarrow 2x^2+3x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+3x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.2.(-2) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.2} & & = \frac{-3+\sqrt25}{2.2} \\ & = \frac{-8}{4} & & = \frac{2}{4} \\ & = -2 & & = \frac{1}{2} \\ \\ V &= \Big\{ -2 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  6. \(-(11+7x)=-4x^2-(47-5x) \\ \Leftrightarrow -11-7x=-4x^2-47+5x \\ \Leftrightarrow 4x^2-12x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-12x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.4.36 & &\\ & = 144-576 & & \\ & = -432 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  7. \((2x-1)(3x-4)-x(-12x-29)=-4\\ \Leftrightarrow 6x^2-8x-3x+4 +12x^2+29x+4=0 \\ \Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
  8. \(x(16x-73)=-49(x+1) \\ \Leftrightarrow 16x^2-73x=-49x-49 \\ \Leftrightarrow 16x^2-24x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-24x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.16.49 & &\\ & = 576-3136 & & \\ & = -2560 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  9. \(\frac{1}{4}x^2+\frac{13}{24}x+\frac{1}{4}=0\\ \Leftrightarrow \color{red}{24.} \left(\frac{1}{4}x^2+\frac{13}{24}x+\frac{1}{4}\right)=0 \color{red}{.24} \\ \Leftrightarrow 6x^2+13x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+13x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.6.6 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.6} & & = \frac{-13+\sqrt25}{2.6} \\ & = \frac{-18}{12} & & = \frac{-8}{12} \\ & = \frac{-3}{2} & & = \frac{-2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{-2}{3} \Big\} & &\end{align} \\ -----------------\)
  10. \(x^2+\frac{5}{2}x+\frac{25}{16}=0\\ \Leftrightarrow \color{red}{16.} \left(x^2+\frac{5}{2}x+\frac{25}{16}\right)=0 \color{red}{.16} \\ \Leftrightarrow 16x^2+40x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+40x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (40)^2-4.16.25 & &\\ & = 1600-1600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-40}{2.16} & & \\ & = -\frac{5}{4} & & \\V &= \Big\{ -\frac{5}{4} \Big\} & &\end{align} \\ -----------------\)
  11. \(\frac{1}{2}x^2+\frac{17}{2}x+35=0\\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+\frac{17}{2}x+35\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2+17x+70=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+17x+70=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.1.70 & &\\ & = 289-280 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt9}{2.1} & & = \frac{-17+\sqrt9}{2.1} \\ & = \frac{-20}{2} & & = \frac{-14}{2} \\ & = -10 & & = -7 \\ \\ V &= \Big\{ -10 ; -7 \Big\} & &\end{align} \\ -----------------\)
  12. \((-x-3)(3x-3)-x(-6x+5)=57\\ \Leftrightarrow -3x^2+3x-9x+9 +6x^2-5x-57=0 \\ \Leftrightarrow 3x^2+7x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+7x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.3.(-48) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.3} & & = \frac{-7+\sqrt625}{2.3} \\ & = \frac{-32}{6} & & = \frac{18}{6} \\ & = \frac{-16}{3} & & = 3 \\ \\ V &= \Big\{ \frac{-16}{3} ; 3 \Big\} & &\end{align} \\ -----------------\)
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