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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-\frac{7}{3}x=-\frac{2}{3}x^2-\frac{49}{24}\)
  2. \(x(x-15)=-25(x+1)\)
  3. \(-(15-29x)=-16x^2-(16-19x)\)
  4. \(x(x-6)=5(x-2)\)
  5. \((2x+3)(-x-1)-x(-3x-7)=-12\)
  6. \(-(8-8x)=-x^2-(-52-12x)\)
  7. \(2x^2-(5x-18)=x(x-16)\)
  8. \(x(x-4)=10(x-4)\)
  9. \(x(9x+1)=5(x-5)\)
  10. \((5x+5)(-3x-1)-x(-16x-23)=-35\)
  11. \(\frac{19}{2}x=-\frac{1}{2}x^2-45\)
  12. \((-4x+4)(2x-5)-x(-26x-5)=-18\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-\frac{7}{3}x=-\frac{2}{3}x^2-\frac{49}{24} \\ \Leftrightarrow \frac{2}{3}x^2-\frac{7}{3}x+\frac{49}{24}=0 \\ \Leftrightarrow \color{red}{24.} \left(\frac{2}{3}x^2-\frac{7}{3}x+\frac{49}{24}\right)=0 \color{red}{.24} \\ \Leftrightarrow 16x^2-56x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-56x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-56)^2-4.16.49 & &\\ & = 3136-3136 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-56)}{2.16} & & \\ & = \frac{7}{4} & & \\V &= \Big\{ \frac{7}{4} \Big\} & &\end{align} \\ -----------------\)
  2. \(x(x-15)=-25(x+1) \\ \Leftrightarrow x^2-15x=-25x-25 \\ \Leftrightarrow x^2+10x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+10x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.1.25 & &\\ & = 100-100 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-10}{2.1} & & \\ & = -5 & & \\V &= \Big\{ -5 \Big\} & &\end{align} \\ -----------------\)
  3. \(-(15-29x)=-16x^2-(16-19x) \\ \Leftrightarrow -15+29x=-16x^2-16+19x \\ \Leftrightarrow 16x^2+10x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+10x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.16.1 & &\\ & = 100-64 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt36}{2.16} & & = \frac{-10+\sqrt36}{2.16} \\ & = \frac{-16}{32} & & = \frac{-4}{32} \\ & = \frac{-1}{2} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
  4. \(x(x-6)=5(x-2) \\ \Leftrightarrow x^2-6x=5x-10 \\ \Leftrightarrow x^2-11x+10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-11)^2-4.1.10 & &\\ & = 121-40 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-11)-\sqrt81}{2.1} & & = \frac{-(-11)+\sqrt81}{2.1} \\ & = \frac{2}{2} & & = \frac{20}{2} \\ & = 1 & & = 10 \\ \\ V &= \Big\{ 1 ; 10 \Big\} & &\end{align} \\ -----------------\)
  5. \((2x+3)(-x-1)-x(-3x-7)=-12\\ \Leftrightarrow -2x^2-2x-3x-3 +3x^2+7x+12=0 \\ \Leftrightarrow x^2+2x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.9 & &\\ & = 4-36 & & \\ & = -32 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  6. \(-(8-8x)=-x^2-(-52-12x) \\ \Leftrightarrow -8+8x=-x^2+52+12x \\ \Leftrightarrow x^2-4x-60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-60) & &\\ & = 16+240 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt256}{2.1} & & = \frac{-(-4)+\sqrt256}{2.1} \\ & = \frac{-12}{2} & & = \frac{20}{2} \\ & = -6 & & = 10 \\ \\ V &= \Big\{ -6 ; 10 \Big\} & &\end{align} \\ -----------------\)
  7. \(2x^2-(5x-18)=x(x-16) \\ \Leftrightarrow 2x^2-5x+18=x^2-16x \\ \Leftrightarrow x^2+11x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.18 & &\\ & = 121-72 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt49}{2.1} & & = \frac{-11+\sqrt49}{2.1} \\ & = \frac{-18}{2} & & = \frac{-4}{2} \\ & = -9 & & = -2 \\ \\ V &= \Big\{ -9 ; -2 \Big\} & &\end{align} \\ -----------------\)
  8. \(x(x-4)=10(x-4) \\ \Leftrightarrow x^2-4x=10x-40 \\ \Leftrightarrow x^2-14x+40=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.40 & &\\ & = 196-160 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-14)-\sqrt36}{2.1} & & = \frac{-(-14)+\sqrt36}{2.1} \\ & = \frac{8}{2} & & = \frac{20}{2} \\ & = 4 & & = 10 \\ \\ V &= \Big\{ 4 ; 10 \Big\} & &\end{align} \\ -----------------\)
  9. \(x(9x+1)=5(x-5) \\ \Leftrightarrow 9x^2+x=5x-25 \\ \Leftrightarrow 9x^2-4x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-4x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.9.25 & &\\ & = 16-900 & & \\ & = -884 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  10. \((5x+5)(-3x-1)-x(-16x-23)=-35\\ \Leftrightarrow -15x^2-5x-15x-5 +16x^2+23x+35=0 \\ \Leftrightarrow x^2+13x+30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+13x+30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.1.30 & &\\ & = 169-120 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt49}{2.1} & & = \frac{-13+\sqrt49}{2.1} \\ & = \frac{-20}{2} & & = \frac{-6}{2} \\ & = -10 & & = -3 \\ \\ V &= \Big\{ -10 ; -3 \Big\} & &\end{align} \\ -----------------\)
  11. \(\frac{19}{2}x=-\frac{1}{2}x^2-45 \\ \Leftrightarrow \frac{1}{2}x^2+\frac{19}{2}x+45=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+\frac{19}{2}x+45\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2+19x+90=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+19x+90=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (19)^2-4.1.90 & &\\ & = 361-360 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-19-\sqrt1}{2.1} & & = \frac{-19+\sqrt1}{2.1} \\ & = \frac{-20}{2} & & = \frac{-18}{2} \\ & = -10 & & = -9 \\ \\ V &= \Big\{ -10 ; -9 \Big\} & &\end{align} \\ -----------------\)
  12. \((-4x+4)(2x-5)-x(-26x-5)=-18\\ \Leftrightarrow -8x^2+20x+8x-20 +26x^2+5x+18=0 \\ \Leftrightarrow 18x^2+5x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+5x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.18.(-2) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.18} & & = \frac{-5+\sqrt169}{2.18} \\ & = \frac{-18}{36} & & = \frac{8}{36} \\ & = \frac{-1}{2} & & = \frac{2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{2}{9} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2025-12-23 05:38:51
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