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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \((-5x+3)(5x+5)-x(-41x-16)=16\)
  2. \(\frac{9}{4}x^2+\frac{13}{4}x+1=0\)
  3. \((2x+1)(-4x-1)-x(-9x-16)=-37\)
  4. \(x(48x+22)=-3(x+1)\)
  5. \(\frac{1}{3}x^2+3x+\frac{121}{12}=0\)
  6. \((-5x-2)(2x-3)-x(-11x+6)=-50\)
  7. \(\frac{1}{2}x^2+\frac{19}{2}x+45=0\)
  8. \(x(24x+13)=6(x+1)\)
  9. \((2x+5)(-x-2)-x(-20x-27)=-12\)
  10. \(x^2+\frac{3}{4}x-\frac{1}{4}=0\)
  11. \(x(x+92)=96(x+1)\)
  12. \((-3x-4)(2x+3)-x(-10x-17)=-13\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \((-5x+3)(5x+5)-x(-41x-16)=16\\ \Leftrightarrow -25x^2-25x+15x+15 +41x^2+16x-16=0 \\ \Leftrightarrow 16x^2+6x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+6x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.16.(-1) & &\\ & = 36+64 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt100}{2.16} & & = \frac{-6+\sqrt100}{2.16} \\ & = \frac{-16}{32} & & = \frac{4}{32} \\ & = \frac{-1}{2} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  2. \(\frac{9}{4}x^2+\frac{13}{4}x+1=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{9}{4}x^2+\frac{13}{4}x+1\right)=0 \color{red}{.4} \\ \Leftrightarrow 9x^2+13x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+13x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.9.4 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.9} & & = \frac{-13+\sqrt25}{2.9} \\ & = \frac{-18}{18} & & = \frac{-8}{18} \\ & = -1 & & = \frac{-4}{9} \\ \\ V &= \Big\{ -1 ; \frac{-4}{9} \Big\} & &\end{align} \\ -----------------\)
  3. \((2x+1)(-4x-1)-x(-9x-16)=-37\\ \Leftrightarrow -8x^2-2x-4x-1 +9x^2+16x+37=0 \\ \Leftrightarrow x^2+13x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+13x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.1.36 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.1} & & = \frac{-13+\sqrt25}{2.1} \\ & = \frac{-18}{2} & & = \frac{-8}{2} \\ & = -9 & & = -4 \\ \\ V &= \Big\{ -9 ; -4 \Big\} & &\end{align} \\ -----------------\)
  4. \(x(48x+22)=-3(x+1) \\ \Leftrightarrow 48x^2+22x=-3x-3 \\ \Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.48.3 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\ & = \frac{-32}{96} & & = \frac{-18}{96} \\ & = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{1}{3}x^2+3x+\frac{121}{12}=0\\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{3}x^2+3x+\frac{121}{12}\right)=0 \color{red}{.12} \\ \Leftrightarrow 4x^2+36x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+36x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (36)^2-4.4.121 & &\\ & = 1296-1936 & & \\ & = -640 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  6. \((-5x-2)(2x-3)-x(-11x+6)=-50\\ \Leftrightarrow -10x^2+15x-4x+6 +11x^2-6x+50=0 \\ \Leftrightarrow x^2+15x+56=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+15x+56=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.1.56 & &\\ & = 225-224 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt1}{2.1} & & = \frac{-15+\sqrt1}{2.1} \\ & = \frac{-16}{2} & & = \frac{-14}{2} \\ & = -8 & & = -7 \\ \\ V &= \Big\{ -8 ; -7 \Big\} & &\end{align} \\ -----------------\)
  7. \(\frac{1}{2}x^2+\frac{19}{2}x+45=0\\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+\frac{19}{2}x+45\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2+19x+90=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+19x+90=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (19)^2-4.1.90 & &\\ & = 361-360 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-19-\sqrt1}{2.1} & & = \frac{-19+\sqrt1}{2.1} \\ & = \frac{-20}{2} & & = \frac{-18}{2} \\ & = -10 & & = -9 \\ \\ V &= \Big\{ -10 ; -9 \Big\} & &\end{align} \\ -----------------\)
  8. \(x(24x+13)=6(x+1) \\ \Leftrightarrow 24x^2+13x=6x+6 \\ \Leftrightarrow 24x^2+7x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+7x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.24.(-6) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.24} & & = \frac{-7+\sqrt625}{2.24} \\ & = \frac{-32}{48} & & = \frac{18}{48} \\ & = \frac{-2}{3} & & = \frac{3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{3}{8} \Big\} & &\end{align} \\ -----------------\)
  9. \((2x+5)(-x-2)-x(-20x-27)=-12\\ \Leftrightarrow -2x^2-4x-5x-10 +20x^2+27x+12=0 \\ \Leftrightarrow 18x^2+13x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+13x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.18.2 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.18} & & = \frac{-13+\sqrt25}{2.18} \\ & = \frac{-18}{36} & & = \frac{-8}{36} \\ & = \frac{-1}{2} & & = \frac{-2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-2}{9} \Big\} & &\end{align} \\ -----------------\)
  10. \(x^2+\frac{3}{4}x-\frac{1}{4}=0\\ \Leftrightarrow \color{red}{4.} \left(x^2+\frac{3}{4}x-\frac{1}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow 4x^2+3x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+3x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.4.(-1) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.4} & & = \frac{-3+\sqrt25}{2.4} \\ & = \frac{-8}{8} & & = \frac{2}{8} \\ & = -1 & & = \frac{1}{4} \\ \\ V &= \Big\{ -1 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
  11. \(x(x+92)=96(x+1) \\ \Leftrightarrow x^2+92x=96x+96 \\ \Leftrightarrow x^2-4x-96=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-96=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-96) & &\\ & = 16+384 & & \\ & = 400 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt400}{2.1} & & = \frac{-(-4)+\sqrt400}{2.1} \\ & = \frac{-16}{2} & & = \frac{24}{2} \\ & = -8 & & = 12 \\ \\ V &= \Big\{ -8 ; 12 \Big\} & &\end{align} \\ -----------------\)
  12. \((-3x-4)(2x+3)-x(-10x-17)=-13\\ \Leftrightarrow -6x^2-9x-8x-12 +10x^2+17x+13=0 \\ \Leftrightarrow 4x^2-4x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-4x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.4.1 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-4)}{2.4} & & \\ & = \frac{1}{2} & & \\V &= \Big\{ \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
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