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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-(15-27x)=-4x^2-(11-12x)\)
  2. \(-(11-12x)=-x^2-(-33-19x)\)
  3. \(4x^2-(2x-48)=3x(x-6)\)
  4. \(-(8-19x)=-9x^2-(24-13x)\)
  5. \(x(48x+22)=-3(x+1)\)
  6. \((4x-4)(-5x-1)-x(-29x-20)=-60\)
  7. \(2x^2-\frac{5}{2}x+\frac{121}{8}=0\)
  8. \((x-4)(5x-4)-x(-67x-13)=14\)
  9. \(-(3-61x)=-16x^2-(52-5x)\)
  10. \(-(2-22x)=-4x^2-(1-19x)\)
  11. \((-x+5)(3x+2)-x(-4x+11)=8\)
  12. \((-4x+3)(3x-1)-x(-13x-7)=-10\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-(15-27x)=-4x^2-(11-12x) \\ \Leftrightarrow -15+27x=-4x^2-11+12x \\ \Leftrightarrow 4x^2+15x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.4.(-4) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\ & = \frac{-32}{8} & & = \frac{2}{8} \\ & = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
  2. \(-(11-12x)=-x^2-(-33-19x) \\ \Leftrightarrow -11+12x=-x^2+33+19x \\ \Leftrightarrow x^2-7x-44=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x-44=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.(-44) & &\\ & = 49+176 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt225}{2.1} & & = \frac{-(-7)+\sqrt225}{2.1} \\ & = \frac{-8}{2} & & = \frac{22}{2} \\ & = -4 & & = 11 \\ \\ V &= \Big\{ -4 ; 11 \Big\} & &\end{align} \\ -----------------\)
  3. \(4x^2-(2x-48)=3x(x-6) \\ \Leftrightarrow 4x^2-2x+48=3x^2-18x \\ \Leftrightarrow x^2+16x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.48 & &\\ & = 256-192 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-16-\sqrt64}{2.1} & & = \frac{-16+\sqrt64}{2.1} \\ & = \frac{-24}{2} & & = \frac{-8}{2} \\ & = -12 & & = -4 \\ \\ V &= \Big\{ -12 ; -4 \Big\} & &\end{align} \\ -----------------\)
  4. \(-(8-19x)=-9x^2-(24-13x) \\ \Leftrightarrow -8+19x=-9x^2-24+13x \\ \Leftrightarrow 9x^2+6x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+6x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.9.16 & &\\ & = 36-576 & & \\ & = -540 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  5. \(x(48x+22)=-3(x+1) \\ \Leftrightarrow 48x^2+22x=-3x-3 \\ \Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.48.3 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\ & = \frac{-32}{96} & & = \frac{-18}{96} \\ & = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
  6. \((4x-4)(-5x-1)-x(-29x-20)=-60\\ \Leftrightarrow -20x^2-4x+20x+4 +29x^2+20x+60=0 \\ \Leftrightarrow 9x^2+20x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+20x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.9.64 & &\\ & = 400-2304 & & \\ & = -1904 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  7. \(2x^2-\frac{5}{2}x+\frac{121}{8}=0\\ \Leftrightarrow \color{red}{8.} \left(2x^2-\frac{5}{2}x+\frac{121}{8}\right)=0 \color{red}{.8} \\ \Leftrightarrow 16x^2-20x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-20x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-20)^2-4.16.121 & &\\ & = 400-7744 & & \\ & = -7344 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  8. \((x-4)(5x-4)-x(-67x-13)=14\\ \Leftrightarrow 5x^2-4x-20x+16 +67x^2+13x-14=0 \\ \Leftrightarrow 72x^2+25x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+25x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.72.2 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.72} & & = \frac{-25+\sqrt49}{2.72} \\ & = \frac{-32}{144} & & = \frac{-18}{144} \\ & = \frac{-2}{9} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
  9. \(-(3-61x)=-16x^2-(52-5x) \\ \Leftrightarrow -3+61x=-16x^2-52+5x \\ \Leftrightarrow 16x^2+56x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+56x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (56)^2-4.16.49 & &\\ & = 3136-3136 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-56}{2.16} & & \\ & = -\frac{7}{4} & & \\V &= \Big\{ -\frac{7}{4} \Big\} & &\end{align} \\ -----------------\)
  10. \(-(2-22x)=-4x^2-(1-19x) \\ \Leftrightarrow -2+22x=-4x^2-1+19x \\ \Leftrightarrow 4x^2+3x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+3x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.4.(-1) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.4} & & = \frac{-3+\sqrt25}{2.4} \\ & = \frac{-8}{8} & & = \frac{2}{8} \\ & = -1 & & = \frac{1}{4} \\ \\ V &= \Big\{ -1 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
  11. \((-x+5)(3x+2)-x(-4x+11)=8\\ \Leftrightarrow -3x^2-2x+15x+10 +4x^2-11x-8=0 \\ \Leftrightarrow x^2-3x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.2 & &\\ & = 9-8 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt1}{2.1} & & = \frac{-(-3)+\sqrt1}{2.1} \\ & = \frac{2}{2} & & = \frac{4}{2} \\ & = 1 & & = 2 \\ \\ V &= \Big\{ 1 ; 2 \Big\} & &\end{align} \\ -----------------\)
  12. \((-4x+3)(3x-1)-x(-13x-7)=-10\\ \Leftrightarrow -12x^2+4x+9x-3 +13x^2+7x+10=0 \\ \Leftrightarrow x^2+8x+7=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x+7=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.7 & &\\ & = 64-28 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-8-\sqrt36}{2.1} & & = \frac{-8+\sqrt36}{2.1} \\ & = \frac{-14}{2} & & = \frac{-2}{2} \\ & = -7 & & = -1 \\ \\ V &= \Big\{ -7 ; -1 \Big\} & &\end{align} \\ -----------------\)
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