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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(3x^2-(14x-9)=2x(x-2)\)
  2. \(\frac{1}{2}x^2+\frac{5}{4}x+\frac{9}{2}=0\)
  3. \(x(x+14)=3(x-10)\)
  4. \(\frac{25}{24}x=-x^2-\frac{1}{4}\)
  5. \(x(x+24)=4(x-25)\)
  6. \(x(4x-30)=2(x-32)\)
  7. \((-5x-2)(-2x+2)-x(-6x-31)=-5\)
  8. \(39x^2-(19x-144)=23x(x-5)\)
  9. \(\frac{17}{5}x=-\frac{1}{5}x^2-12\)
  10. \(2x^2-(6x-20)=x(x-15)\)
  11. \(-(4-22x)=-48x^2-(1-15x)\)
  12. \((-3x-4)(-x-1)-x(2x-7)=-44\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(3x^2-(14x-9)=2x(x-2) \\ \Leftrightarrow 3x^2-14x+9=2x^2-4x \\ \Leftrightarrow x^2-10x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.9 & &\\ & = 100-36 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-10)-\sqrt64}{2.1} & & = \frac{-(-10)+\sqrt64}{2.1} \\ & = \frac{2}{2} & & = \frac{18}{2} \\ & = 1 & & = 9 \\ \\ V &= \Big\{ 1 ; 9 \Big\} & &\end{align} \\ -----------------\)
  2. \(\frac{1}{2}x^2+\frac{5}{4}x+\frac{9}{2}=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{2}x^2+\frac{5}{4}x+\frac{9}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow 4x^2+10x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+10x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.4.36 & &\\ & = 100-576 & & \\ & = -476 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  3. \(x(x+14)=3(x-10) \\ \Leftrightarrow x^2+14x=3x-30 \\ \Leftrightarrow x^2+11x+30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.30 & &\\ & = 121-120 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt1}{2.1} & & = \frac{-11+\sqrt1}{2.1} \\ & = \frac{-12}{2} & & = \frac{-10}{2} \\ & = -6 & & = -5 \\ \\ V &= \Big\{ -6 ; -5 \Big\} & &\end{align} \\ -----------------\)
  4. \(\frac{25}{24}x=-x^2-\frac{1}{4} \\ \Leftrightarrow x^2+\frac{25}{24}x+\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{24.} \left(x^2+\frac{25}{24}x+\frac{1}{4}\right)=0 \color{red}{.24} \\ \Leftrightarrow 24x^2+25x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+25x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.24.6 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.24} & & = \frac{-25+\sqrt49}{2.24} \\ & = \frac{-32}{48} & & = \frac{-18}{48} \\ & = \frac{-2}{3} & & = \frac{-3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{-3}{8} \Big\} & &\end{align} \\ -----------------\)
  5. \(x(x+24)=4(x-25) \\ \Leftrightarrow x^2+24x=4x-100 \\ \Leftrightarrow x^2+20x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+20x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.1.100 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.1} & & \\ & = -10 & & \\V &= \Big\{ -10 \Big\} & &\end{align} \\ -----------------\)
  6. \(x(4x-30)=2(x-32) \\ \Leftrightarrow 4x^2-30x=2x-64 \\ \Leftrightarrow 4x^2-32x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-32x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-32)^2-4.4.64 & &\\ & = 1024-1024 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-32)}{2.4} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
  7. \((-5x-2)(-2x+2)-x(-6x-31)=-5\\ \Leftrightarrow 10x^2-10x+4x-4 +6x^2+31x+5=0 \\ \Leftrightarrow 16x^2+17x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+17x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.16.1 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.16} & & = \frac{-17+\sqrt225}{2.16} \\ & = \frac{-32}{32} & & = \frac{-2}{32} \\ & = -1 & & = \frac{-1}{16} \\ \\ V &= \Big\{ -1 ; \frac{-1}{16} \Big\} & &\end{align} \\ -----------------\)
  8. \(39x^2-(19x-144)=23x(x-5) \\ \Leftrightarrow 39x^2-19x+144=23x^2-115x \\ \Leftrightarrow 16x^2+96x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+96x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (96)^2-4.16.144 & &\\ & = 9216-9216 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-96}{2.16} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
  9. \(\frac{17}{5}x=-\frac{1}{5}x^2-12 \\ \Leftrightarrow \frac{1}{5}x^2+\frac{17}{5}x+12=0 \\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+\frac{17}{5}x+12\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2+17x+60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+17x+60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.1.60 & &\\ & = 289-240 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt49}{2.1} & & = \frac{-17+\sqrt49}{2.1} \\ & = \frac{-24}{2} & & = \frac{-10}{2} \\ & = -12 & & = -5 \\ \\ V &= \Big\{ -12 ; -5 \Big\} & &\end{align} \\ -----------------\)
  10. \(2x^2-(6x-20)=x(x-15) \\ \Leftrightarrow 2x^2-6x+20=x^2-15x \\ \Leftrightarrow x^2+9x+20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+9x+20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (9)^2-4.1.20 & &\\ & = 81-80 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-9-\sqrt1}{2.1} & & = \frac{-9+\sqrt1}{2.1} \\ & = \frac{-10}{2} & & = \frac{-8}{2} \\ & = -5 & & = -4 \\ \\ V &= \Big\{ -5 ; -4 \Big\} & &\end{align} \\ -----------------\)
  11. \(-(4-22x)=-48x^2-(1-15x) \\ \Leftrightarrow -4+22x=-48x^2-1+15x \\ \Leftrightarrow 48x^2+7x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
  12. \((-3x-4)(-x-1)-x(2x-7)=-44\\ \Leftrightarrow 3x^2+3x+4x+4 -2x^2+7x+44=0 \\ \Leftrightarrow x^2+14x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.48 & &\\ & = 196-192 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-14-\sqrt4}{2.1} & & = \frac{-14+\sqrt4}{2.1} \\ & = \frac{-16}{2} & & = \frac{-12}{2} \\ & = -8 & & = -6 \\ \\ V &= \Big\{ -8 ; -6 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-06-08 15:18:32
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