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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \((-5x+1)(-x-4)-x(x+11)=-5\)
  2. \(5x^2-(8x-144)=x(x+40)\)
  3. \(-(5+4x)=-x^2-(12-4x)\)
  4. \(-(8-15x)=-x^2-(6-16x)\)
  5. \(\frac{7}{30}x=-\frac{4}{5}x^2+\frac{1}{5}\)
  6. \(\frac{9}{5}x^2+\frac{38}{5}x+\frac{121}{5}=0\)
  7. \(2x^2-(10x-96)=x(x-30)\)
  8. \(3x^2-(19x+72)=x(x-26)\)
  9. \(-\frac{1}{2}x=-\frac{1}{8}x^2+12\)
  10. \(2x^2-(13x-120)=x(x+9)\)
  11. \(-(9-39x)=-48x^2-(12-14x)\)
  12. \(-(12+35x)=-16x^2-(48-5x)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \((-5x+1)(-x-4)-x(x+11)=-5\\ \Leftrightarrow 5x^2+20x-x-4 -x^2-11x+5=0 \\ \Leftrightarrow 4x^2+5x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.1 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.4} & & = \frac{-5+\sqrt9}{2.4} \\ & = \frac{-8}{8} & & = \frac{-2}{8} \\ & = -1 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -1 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  2. \(5x^2-(8x-144)=x(x+40) \\ \Leftrightarrow 5x^2-8x+144=x^2+40x \\ \Leftrightarrow 4x^2-48x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-48x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-48)^2-4.4.144 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-48)}{2.4} & & \\ & = 6 & & \\V &= \Big\{ 6 \Big\} & &\end{align} \\ -----------------\)
  3. \(-(5+4x)=-x^2-(12-4x) \\ \Leftrightarrow -5-4x=-x^2-12+4x \\ \Leftrightarrow x^2-8x+7=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+7=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.7 & &\\ & = 64-28 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt36}{2.1} & & = \frac{-(-8)+\sqrt36}{2.1} \\ & = \frac{2}{2} & & = \frac{14}{2} \\ & = 1 & & = 7 \\ \\ V &= \Big\{ 1 ; 7 \Big\} & &\end{align} \\ -----------------\)
  4. \(-(8-15x)=-x^2-(6-16x) \\ \Leftrightarrow -8+15x=-x^2-6+16x \\ \Leftrightarrow x^2-x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-2) & &\\ & = 1+8 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt9}{2.1} & & = \frac{-(-1)+\sqrt9}{2.1} \\ & = \frac{-2}{2} & & = \frac{4}{2} \\ & = -1 & & = 2 \\ \\ V &= \Big\{ -1 ; 2 \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{7}{30}x=-\frac{4}{5}x^2+\frac{1}{5} \\ \Leftrightarrow \frac{4}{5}x^2+\frac{7}{30}x-\frac{1}{5}=0 \\ \Leftrightarrow \color{red}{30.} \left(\frac{4}{5}x^2+\frac{7}{30}x-\frac{1}{5}\right)=0 \color{red}{.30} \\ \Leftrightarrow 24x^2+7x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+7x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.24.(-6) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.24} & & = \frac{-7+\sqrt625}{2.24} \\ & = \frac{-32}{48} & & = \frac{18}{48} \\ & = \frac{-2}{3} & & = \frac{3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{3}{8} \Big\} & &\end{align} \\ -----------------\)
  6. \(\frac{9}{5}x^2+\frac{38}{5}x+\frac{121}{5}=0\\ \Leftrightarrow \color{red}{5.} \left(\frac{9}{5}x^2+\frac{38}{5}x+\frac{121}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow 9x^2+38x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+38x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (38)^2-4.9.121 & &\\ & = 1444-4356 & & \\ & = -2912 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  7. \(2x^2-(10x-96)=x(x-30) \\ \Leftrightarrow 2x^2-10x+96=x^2-30x \\ \Leftrightarrow x^2+20x+96=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+20x+96=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.1.96 & &\\ & = 400-384 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-20-\sqrt16}{2.1} & & = \frac{-20+\sqrt16}{2.1} \\ & = \frac{-24}{2} & & = \frac{-16}{2} \\ & = -12 & & = -8 \\ \\ V &= \Big\{ -12 ; -8 \Big\} & &\end{align} \\ -----------------\)
  8. \(3x^2-(19x+72)=x(x-26) \\ \Leftrightarrow 3x^2-19x-72=x^2-26x \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
  9. \(-\frac{1}{2}x=-\frac{1}{8}x^2+12 \\ \Leftrightarrow \frac{1}{8}x^2-\frac{1}{2}x-12=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2-\frac{1}{2}x-12\right)=0 \color{red}{.8} \\ \Leftrightarrow x^2-4x-96=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-96=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-96) & &\\ & = 16+384 & & \\ & = 400 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt400}{2.1} & & = \frac{-(-4)+\sqrt400}{2.1} \\ & = \frac{-16}{2} & & = \frac{24}{2} \\ & = -8 & & = 12 \\ \\ V &= \Big\{ -8 ; 12 \Big\} & &\end{align} \\ -----------------\)
  10. \(2x^2-(13x-120)=x(x+9) \\ \Leftrightarrow 2x^2-13x+120=x^2+9x \\ \Leftrightarrow x^2-22x+120=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-22x+120=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.1.120 & &\\ & = 484-480 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-22)-\sqrt4}{2.1} & & = \frac{-(-22)+\sqrt4}{2.1} \\ & = \frac{20}{2} & & = \frac{24}{2} \\ & = 10 & & = 12 \\ \\ V &= \Big\{ 10 ; 12 \Big\} & &\end{align} \\ -----------------\)
  11. \(-(9-39x)=-48x^2-(12-14x) \\ \Leftrightarrow -9+39x=-48x^2-12+14x \\ \Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.48.3 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\ & = \frac{-32}{96} & & = \frac{-18}{96} \\ & = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
  12. \(-(12+35x)=-16x^2-(48-5x) \\ \Leftrightarrow -12-35x=-16x^2-48+5x \\ \Leftrightarrow 16x^2-40x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-40x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-40)^2-4.16.36 & &\\ & = 1600-2304 & & \\ & = -704 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
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