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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(2x^2-(6x-48)=x(x-22)\)
  2. \((2x+5)(-x+3)-x(-6x-4)=-21\)
  3. \(23x^2-(12x-4)=7x(x-4)\)
  4. \((4x-1)(-x+4)-x(-5x-2)=-52\)
  5. \(2x^2-(7x+80)=x(x-5)\)
  6. \(-\frac{21}{32}x=-\frac{1}{4}x^2-\frac{9}{4}\)
  7. \(\frac{4}{5}x^2+\frac{3}{2}x-\frac{1}{5}=0\)
  8. \(x(8x+25)=18(x+1)\)
  9. \((x+1)(5x+3)-x(4x+5)=135\)
  10. \((5x+2)(x+2)-x(4x+19)=10\)
  11. \((-5x-5)(4x+1)-x(-21x-14)=-9\)
  12. \((-5x+4)(2x-5)-x(-11x+1)=12\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(2x^2-(6x-48)=x(x-22) \\ \Leftrightarrow 2x^2-6x+48=x^2-22x \\ \Leftrightarrow x^2+16x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.48 & &\\ & = 256-192 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-16-\sqrt64}{2.1} & & = \frac{-16+\sqrt64}{2.1} \\ & = \frac{-24}{2} & & = \frac{-8}{2} \\ & = -12 & & = -4 \\ \\ V &= \Big\{ -12 ; -4 \Big\} & &\end{align} \\ -----------------\)
  2. \((2x+5)(-x+3)-x(-6x-4)=-21\\ \Leftrightarrow -2x^2+6x-5x+15 +6x^2+4x+21=0 \\ \Leftrightarrow 4x^2+25x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+25x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.4.36 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.4} & & = \frac{-25+\sqrt49}{2.4} \\ & = \frac{-32}{8} & & = \frac{-18}{8} \\ & = -4 & & = \frac{-9}{4} \\ \\ V &= \Big\{ -4 ; \frac{-9}{4} \Big\} & &\end{align} \\ -----------------\)
  3. \(23x^2-(12x-4)=7x(x-4) \\ \Leftrightarrow 23x^2-12x+4=7x^2-28x \\ \Leftrightarrow 16x^2+16x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+16x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.16.4 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.16} & & \\ & = -\frac{1}{2} & & \\V &= \Big\{ -\frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  4. \((4x-1)(-x+4)-x(-5x-2)=-52\\ \Leftrightarrow -4x^2+16x+x-4 +5x^2+2x+52=0 \\ \Leftrightarrow x^2+14x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.48 & &\\ & = 196-192 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-14-\sqrt4}{2.1} & & = \frac{-14+\sqrt4}{2.1} \\ & = \frac{-16}{2} & & = \frac{-12}{2} \\ & = -8 & & = -6 \\ \\ V &= \Big\{ -8 ; -6 \Big\} & &\end{align} \\ -----------------\)
  5. \(2x^2-(7x+80)=x(x-5) \\ \Leftrightarrow 2x^2-7x-80=x^2-5x \\ \Leftrightarrow x^2-2x-80=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-80=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-80) & &\\ & = 4+320 & & \\ & = 324 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt324}{2.1} & & = \frac{-(-2)+\sqrt324}{2.1} \\ & = \frac{-16}{2} & & = \frac{20}{2} \\ & = -8 & & = 10 \\ \\ V &= \Big\{ -8 ; 10 \Big\} & &\end{align} \\ -----------------\)
  6. \(-\frac{21}{32}x=-\frac{1}{4}x^2-\frac{9}{4} \\ \Leftrightarrow \frac{1}{4}x^2-\frac{21}{32}x+\frac{9}{4}=0 \\ \Leftrightarrow \color{red}{32.} \left(\frac{1}{4}x^2-\frac{21}{32}x+\frac{9}{4}\right)=0 \color{red}{.32} \\ \Leftrightarrow 16x^2-42x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-42x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-42)^2-4.16.144 & &\\ & = 1764-9216 & & \\ & = -7452 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  7. \(\frac{4}{5}x^2+\frac{3}{2}x-\frac{1}{5}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{4}{5}x^2+\frac{3}{2}x-\frac{1}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 8x^2+15x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+15x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.8.(-2) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.8} & & = \frac{-15+\sqrt289}{2.8} \\ & = \frac{-32}{16} & & = \frac{2}{16} \\ & = -2 & & = \frac{1}{8} \\ \\ V &= \Big\{ -2 ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  8. \(x(8x+25)=18(x+1) \\ \Leftrightarrow 8x^2+25x=18x+18 \\ \Leftrightarrow 8x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.8.(-18) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\ & = \frac{-32}{16} & & = \frac{18}{16} \\ & = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)
  9. \((x+1)(5x+3)-x(4x+5)=135\\ \Leftrightarrow 5x^2+3x+5x+3 -4x^2-5x-135=0 \\ \Leftrightarrow x^2+x-132=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-132=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-132) & &\\ & = 1+528 & & \\ & = 529 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt529}{2.1} & & = \frac{-1+\sqrt529}{2.1} \\ & = \frac{-24}{2} & & = \frac{22}{2} \\ & = -12 & & = 11 \\ \\ V &= \Big\{ -12 ; 11 \Big\} & &\end{align} \\ -----------------\)
  10. \((5x+2)(x+2)-x(4x+19)=10\\ \Leftrightarrow 5x^2+10x+2x+4 -4x^2-19x-10=0 \\ \Leftrightarrow x^2-5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-6) & &\\ & = 25+24 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt49}{2.1} & & = \frac{-(-5)+\sqrt49}{2.1} \\ & = \frac{-2}{2} & & = \frac{12}{2} \\ & = -1 & & = 6 \\ \\ V &= \Big\{ -1 ; 6 \Big\} & &\end{align} \\ -----------------\)
  11. \((-5x-5)(4x+1)-x(-21x-14)=-9\\ \Leftrightarrow -20x^2-5x-20x-5 +21x^2+14x+9=0 \\ \Leftrightarrow x^2+4x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.1} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
  12. \((-5x+4)(2x-5)-x(-11x+1)=12\\ \Leftrightarrow -10x^2+25x+8x-20 +11x^2-x-12=0 \\ \Leftrightarrow x^2+4x-32=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-32=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-32) & &\\ & = 16+128 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt144}{2.1} & & = \frac{-4+\sqrt144}{2.1} \\ & = \frac{-16}{2} & & = \frac{8}{2} \\ & = -8 & & = 4 \\ \\ V &= \Big\{ -8 ; 4 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2025-12-13 00:07:33
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