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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-(4+17x)=-x^2-(125-5x)\)
  2. \(-(6-6x)=-x^2-(-90-2x)\)
  3. \(-(8-2x)=-x^2-(24-10x)\)
  4. \(\frac{1}{2}x^2-4x-\frac{33}{2}=0\)
  5. \(-(8+6x)=-x^2-(26-3x)\)
  6. \(x(6x+31)=24(x+1)\)
  7. \(\frac{15}{2}x=-8x^2+\frac{1}{2}\)
  8. \(\frac{1}{5}x^2+\frac{3}{4}x-\frac{1}{5}=0\)
  9. \(17x^2-(12x-144)=x(x+12)\)
  10. \((4x+3)(-5x-1)-x(-21x-15)=-19\)
  11. \((-4x-3)(3x-3)-x(-30x-4)=1\)
  12. \(-(7-22x)=-2x^2-(9-17x)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-(4+17x)=-x^2-(125-5x) \\ \Leftrightarrow -4-17x=-x^2-125+5x \\ \Leftrightarrow x^2-22x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-22x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.1.121 & &\\ & = 484-484 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-22)}{2.1} & & \\ & = 11 & & \\V &= \Big\{ 11 \Big\} & &\end{align} \\ -----------------\)
  2. \(-(6-6x)=-x^2-(-90-2x) \\ \Leftrightarrow -6+6x=-x^2+90+2x \\ \Leftrightarrow x^2+4x-96=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-96=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-96) & &\\ & = 16+384 & & \\ & = 400 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt400}{2.1} & & = \frac{-4+\sqrt400}{2.1} \\ & = \frac{-24}{2} & & = \frac{16}{2} \\ & = -12 & & = 8 \\ \\ V &= \Big\{ -12 ; 8 \Big\} & &\end{align} \\ -----------------\)
  3. \(-(8-2x)=-x^2-(24-10x) \\ \Leftrightarrow -8+2x=-x^2-24+10x \\ \Leftrightarrow x^2-8x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.16 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-8)}{2.1} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
  4. \(\frac{1}{2}x^2-4x-\frac{33}{2}=0\\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-4x-\frac{33}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2-8x-33=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-33=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-33) & &\\ & = 64+132 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt196}{2.1} & & = \frac{-(-8)+\sqrt196}{2.1} \\ & = \frac{-6}{2} & & = \frac{22}{2} \\ & = -3 & & = 11 \\ \\ V &= \Big\{ -3 ; 11 \Big\} & &\end{align} \\ -----------------\)
  5. \(-(8+6x)=-x^2-(26-3x) \\ \Leftrightarrow -8-6x=-x^2-26+3x \\ \Leftrightarrow x^2-9x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-9)^2-4.1.18 & &\\ & = 81-72 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-9)-\sqrt9}{2.1} & & = \frac{-(-9)+\sqrt9}{2.1} \\ & = \frac{6}{2} & & = \frac{12}{2} \\ & = 3 & & = 6 \\ \\ V &= \Big\{ 3 ; 6 \Big\} & &\end{align} \\ -----------------\)
  6. \(x(6x+31)=24(x+1) \\ \Leftrightarrow 6x^2+31x=24x+24 \\ \Leftrightarrow 6x^2+7x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+7x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.6.(-24) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.6} & & = \frac{-7+\sqrt625}{2.6} \\ & = \frac{-32}{12} & & = \frac{18}{12} \\ & = \frac{-8}{3} & & = \frac{3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
  7. \(\frac{15}{2}x=-8x^2+\frac{1}{2} \\ \Leftrightarrow 8x^2+\frac{15}{2}x-\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{2.} \left(8x^2+\frac{15}{2}x-\frac{1}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow 16x^2+15x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+15x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.16.(-1) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.16} & & = \frac{-15+\sqrt289}{2.16} \\ & = \frac{-32}{32} & & = \frac{2}{32} \\ & = -1 & & = \frac{1}{16} \\ \\ V &= \Big\{ -1 ; \frac{1}{16} \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{1}{5}x^2+\frac{3}{4}x-\frac{1}{5}=0\\ \Leftrightarrow \color{red}{20.} \left(\frac{1}{5}x^2+\frac{3}{4}x-\frac{1}{5}\right)=0 \color{red}{.20} \\ \Leftrightarrow 4x^2+15x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.4.(-4) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\ & = \frac{-32}{8} & & = \frac{2}{8} \\ & = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
  9. \(17x^2-(12x-144)=x(x+12) \\ \Leftrightarrow 17x^2-12x+144=x^2+12x \\ \Leftrightarrow 16x^2-24x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-24x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.16.144 & &\\ & = 576-9216 & & \\ & = -8640 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  10. \((4x+3)(-5x-1)-x(-21x-15)=-19\\ \Leftrightarrow -20x^2-4x-15x-3 +21x^2+15x+19=0 \\ \Leftrightarrow x^2+8x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.16 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-8}{2.1} & & \\ & = -4 & & \\V &= \Big\{ -4 \Big\} & &\end{align} \\ -----------------\)
  11. \((-4x-3)(3x-3)-x(-30x-4)=1\\ \Leftrightarrow -12x^2+12x-9x+9 +30x^2+4x-1=0 \\ \Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
  12. \(-(7-22x)=-2x^2-(9-17x) \\ \Leftrightarrow -7+22x=-2x^2-9+17x \\ \Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-04-12 04:40:18
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