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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \((-3x+3)(5x+4)-x(-19x-48)=-132\)
  2. \(-(5-13x)=-x^2-(-58-15x)\)
  3. \((-4x+4)(-2x+4)-x(-10x-7)=24\)
  4. \((4x-2)(-5x-4)-x(-21x-21)=-22\)
  5. \(\frac{1}{12}x^2-\frac{1}{2}x-\frac{9}{4}=0\)
  6. \(x(16x-39)=9(x-4)\)
  7. \(x(4x+1)=-(x+1)\)
  8. \(17x^2-(13x-1)=x(x-30)\)
  9. \(x(x+21)=7(x-7)\)
  10. \(x(x+14)=24(x+1)\)
  11. \(11x^2-(18x+1)=7x(x-3)\)
  12. \(x(2x+3)=-2(x+1)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \((-3x+3)(5x+4)-x(-19x-48)=-132\\ \Leftrightarrow -15x^2-12x+15x+12 +19x^2+48x+132=0 \\ \Leftrightarrow 4x^2+48x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+48x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (48)^2-4.4.144 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-48}{2.4} & & \\ & = -6 & & \\V &= \Big\{ -6 \Big\} & &\end{align} \\ -----------------\)
  2. \(-(5-13x)=-x^2-(-58-15x) \\ \Leftrightarrow -5+13x=-x^2+58+15x \\ \Leftrightarrow x^2-2x-63=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-63=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-63) & &\\ & = 4+252 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt256}{2.1} & & = \frac{-(-2)+\sqrt256}{2.1} \\ & = \frac{-14}{2} & & = \frac{18}{2} \\ & = -7 & & = 9 \\ \\ V &= \Big\{ -7 ; 9 \Big\} & &\end{align} \\ -----------------\)
  3. \((-4x+4)(-2x+4)-x(-10x-7)=24\\ \Leftrightarrow 8x^2-16x-8x+16 +10x^2+7x-24=0 \\ \Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  4. \((4x-2)(-5x-4)-x(-21x-21)=-22\\ \Leftrightarrow -20x^2-16x+10x+8 +21x^2+21x+22=0 \\ \Leftrightarrow x^2+13x+30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+13x+30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.1.30 & &\\ & = 169-120 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt49}{2.1} & & = \frac{-13+\sqrt49}{2.1} \\ & = \frac{-20}{2} & & = \frac{-6}{2} \\ & = -10 & & = -3 \\ \\ V &= \Big\{ -10 ; -3 \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{1}{12}x^2-\frac{1}{2}x-\frac{9}{4}=0\\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{12}x^2-\frac{1}{2}x-\frac{9}{4}\right)=0 \color{red}{.12} \\ \Leftrightarrow x^2-6x-27=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-27=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.(-27) & &\\ & = 36+108 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt144}{2.1} & & = \frac{-(-6)+\sqrt144}{2.1} \\ & = \frac{-6}{2} & & = \frac{18}{2} \\ & = -3 & & = 9 \\ \\ V &= \Big\{ -3 ; 9 \Big\} & &\end{align} \\ -----------------\)
  6. \(x(16x-39)=9(x-4) \\ \Leftrightarrow 16x^2-39x=9x-36 \\ \Leftrightarrow 16x^2-48x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-48x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-48)^2-4.16.36 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-48)}{2.16} & & \\ & = \frac{3}{2} & & \\V &= \Big\{ \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
  7. \(x(4x+1)=-(x+1) \\ \Leftrightarrow 4x^2+x=-x-1 \\ \Leftrightarrow 4x^2+2x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+2x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.4.1 & &\\ & = 4-16 & & \\ & = -12 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  8. \(17x^2-(13x-1)=x(x-30) \\ \Leftrightarrow 17x^2-13x+1=x^2-30x \\ \Leftrightarrow 16x^2+17x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+17x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.16.1 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.16} & & = \frac{-17+\sqrt225}{2.16} \\ & = \frac{-32}{32} & & = \frac{-2}{32} \\ & = -1 & & = \frac{-1}{16} \\ \\ V &= \Big\{ -1 ; \frac{-1}{16} \Big\} & &\end{align} \\ -----------------\)
  9. \(x(x+21)=7(x-7) \\ \Leftrightarrow x^2+21x=7x-49 \\ \Leftrightarrow x^2+14x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.49 & &\\ & = 196-196 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-14}{2.1} & & \\ & = -7 & & \\V &= \Big\{ -7 \Big\} & &\end{align} \\ -----------------\)
  10. \(x(x+14)=24(x+1) \\ \Leftrightarrow x^2+14x=24x+24 \\ \Leftrightarrow x^2-10x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.(-24) & &\\ & = 100+96 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-10)-\sqrt196}{2.1} & & = \frac{-(-10)+\sqrt196}{2.1} \\ & = \frac{-4}{2} & & = \frac{24}{2} \\ & = -2 & & = 12 \\ \\ V &= \Big\{ -2 ; 12 \Big\} & &\end{align} \\ -----------------\)
  11. \(11x^2-(18x+1)=7x(x-3) \\ \Leftrightarrow 11x^2-18x-1=7x^2-21x \\ \Leftrightarrow 4x^2+3x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+3x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.4.(-1) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.4} & & = \frac{-3+\sqrt25}{2.4} \\ & = \frac{-8}{8} & & = \frac{2}{8} \\ & = -1 & & = \frac{1}{4} \\ \\ V &= \Big\{ -1 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
  12. \(x(2x+3)=-2(x+1) \\ \Leftrightarrow 2x^2+3x=-2x-2 \\ \Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
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