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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{15}{4}x=-\frac{1}{4}x^2-14\)
  2. \((-3x-4)(3x-5)-x(-18x+33)=19\)
  3. \(-(4-38x)=-18x^2-(12-13x)\)
  4. \(x(x+8)=10(x-10)\)
  5. \(-(9+5x)=-16x^2-(34-3x)\)
  6. \(x(x+21)=27(x+1)\)
  7. \((4x-4)(2x+1)-x(7x+8)=44\)
  8. \(-\frac{5}{2}x=-\frac{1}{2}x^2-3\)
  9. \(x(2x+23)=18(x+1)\)
  10. \(2x^2-(5x+48)=x(x-7)\)
  11. \(x(x-11)=11(x-11)\)
  12. \(-\frac{1}{5}x=-\frac{1}{25}x^2+2\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{15}{4}x=-\frac{1}{4}x^2-14 \\ \Leftrightarrow \frac{1}{4}x^2+\frac{15}{4}x+14=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{15}{4}x+14\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+15x+56=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+15x+56=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.1.56 & &\\ & = 225-224 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt1}{2.1} & & = \frac{-15+\sqrt1}{2.1} \\ & = \frac{-16}{2} & & = \frac{-14}{2} \\ & = -8 & & = -7 \\ \\ V &= \Big\{ -8 ; -7 \Big\} & &\end{align} \\ -----------------\)
  2. \((-3x-4)(3x-5)-x(-18x+33)=19\\ \Leftrightarrow -9x^2+15x-12x+20 +18x^2-33x-19=0 \\ \Leftrightarrow 9x^2+2x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+2x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.9.1 & &\\ & = 4-36 & & \\ & = -32 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  3. \(-(4-38x)=-18x^2-(12-13x) \\ \Leftrightarrow -4+38x=-18x^2-12+13x \\ \Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
  4. \(x(x+8)=10(x-10) \\ \Leftrightarrow x^2+8x=10x-100 \\ \Leftrightarrow x^2-2x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.100 & &\\ & = 4-400 & & \\ & = -396 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  5. \(-(9+5x)=-16x^2-(34-3x) \\ \Leftrightarrow -9-5x=-16x^2-34+3x \\ \Leftrightarrow 16x^2-8x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-8x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.16.25 & &\\ & = 64-1600 & & \\ & = -1536 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  6. \(x(x+21)=27(x+1) \\ \Leftrightarrow x^2+21x=27x+27 \\ \Leftrightarrow x^2-6x-27=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-27=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.(-27) & &\\ & = 36+108 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt144}{2.1} & & = \frac{-(-6)+\sqrt144}{2.1} \\ & = \frac{-6}{2} & & = \frac{18}{2} \\ & = -3 & & = 9 \\ \\ V &= \Big\{ -3 ; 9 \Big\} & &\end{align} \\ -----------------\)
  7. \((4x-4)(2x+1)-x(7x+8)=44\\ \Leftrightarrow 8x^2+4x-8x-4 -7x^2-8x-44=0 \\ \Leftrightarrow x^2-8x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-48) & &\\ & = 64+192 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt256}{2.1} & & = \frac{-(-8)+\sqrt256}{2.1} \\ & = \frac{-8}{2} & & = \frac{24}{2} \\ & = -4 & & = 12 \\ \\ V &= \Big\{ -4 ; 12 \Big\} & &\end{align} \\ -----------------\)
  8. \(-\frac{5}{2}x=-\frac{1}{2}x^2-3 \\ \Leftrightarrow \frac{1}{2}x^2-\frac{5}{2}x+3=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-\frac{5}{2}x+3\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2-5x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.6 & &\\ & = 25-24 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt1}{2.1} & & = \frac{-(-5)+\sqrt1}{2.1} \\ & = \frac{4}{2} & & = \frac{6}{2} \\ & = 2 & & = 3 \\ \\ V &= \Big\{ 2 ; 3 \Big\} & &\end{align} \\ -----------------\)
  9. \(x(2x+23)=18(x+1) \\ \Leftrightarrow 2x^2+23x=18x+18 \\ \Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.(-18) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\ & = \frac{-18}{4} & & = \frac{8}{4} \\ & = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
  10. \(2x^2-(5x+48)=x(x-7) \\ \Leftrightarrow 2x^2-5x-48=x^2-7x \\ \Leftrightarrow x^2+2x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-48) & &\\ & = 4+192 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt196}{2.1} & & = \frac{-2+\sqrt196}{2.1} \\ & = \frac{-16}{2} & & = \frac{12}{2} \\ & = -8 & & = 6 \\ \\ V &= \Big\{ -8 ; 6 \Big\} & &\end{align} \\ -----------------\)
  11. \(x(x-11)=11(x-11) \\ \Leftrightarrow x^2-11x=11x-121 \\ \Leftrightarrow x^2-22x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-22x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.1.121 & &\\ & = 484-484 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-22)}{2.1} & & \\ & = 11 & & \\V &= \Big\{ 11 \Big\} & &\end{align} \\ -----------------\)
  12. \(-\frac{1}{5}x=-\frac{1}{25}x^2+2 \\ \Leftrightarrow \frac{1}{25}x^2-\frac{1}{5}x-2=0 \\ \Leftrightarrow \color{red}{25.} \left(\frac{1}{25}x^2-\frac{1}{5}x-2\right)=0 \color{red}{.25} \\ \Leftrightarrow x^2-5x-50=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-50=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-50) & &\\ & = 25+200 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt225}{2.1} & & = \frac{-(-5)+\sqrt225}{2.1} \\ & = \frac{-10}{2} & & = \frac{20}{2} \\ & = -5 & & = 10 \\ \\ V &= \Big\{ -5 ; 10 \Big\} & &\end{align} \\ -----------------\)
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