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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-\frac{1}{2}x=-\frac{1}{16}x^2-1\)
  2. \((-2x+1)(3x-5)-x(-7x-11)=-65\)
  3. \(x(x-5)=2(x-3)\)
  4. \((-2x+1)(-x+1)-x(x+5)=-8\)
  5. \(x(x-7)=3(x-3)\)
  6. \(-(8-63x)=-9x^2-(89-17x)\)
  7. \(-\frac{5}{4}x=-\frac{1}{4}x^2-1\)
  8. \(x(x+11)=12(x+1)\)
  9. \(x(3x+29)=4(x-12)\)
  10. \(\frac{13}{3}x=-\frac{4}{3}x^2-3\)
  11. \((-5x-2)(-x-5)-x(-31x+10)=6\)
  12. \((5x+4)(3x-2)-x(14x-21)=2\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-\frac{1}{2}x=-\frac{1}{16}x^2-1 \\ \Leftrightarrow \frac{1}{16}x^2-\frac{1}{2}x+1=0 \\ \Leftrightarrow \color{red}{16.} \left(\frac{1}{16}x^2-\frac{1}{2}x+1\right)=0 \color{red}{.16} \\ \Leftrightarrow 4x^2-32x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-32x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-32)^2-4.4.64 & &\\ & = 1024-1024 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-32)}{2.4} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
  2. \((-2x+1)(3x-5)-x(-7x-11)=-65\\ \Leftrightarrow -6x^2+10x+3x-5 +7x^2+11x+65=0 \\ \Leftrightarrow x^2+16x+60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.60 & &\\ & = 256-240 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-16-\sqrt16}{2.1} & & = \frac{-16+\sqrt16}{2.1} \\ & = \frac{-20}{2} & & = \frac{-12}{2} \\ & = -10 & & = -6 \\ \\ V &= \Big\{ -10 ; -6 \Big\} & &\end{align} \\ -----------------\)
  3. \(x(x-5)=2(x-3) \\ \Leftrightarrow x^2-5x=2x-6 \\ \Leftrightarrow x^2-7x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.6 & &\\ & = 49-24 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt25}{2.1} & & = \frac{-(-7)+\sqrt25}{2.1} \\ & = \frac{2}{2} & & = \frac{12}{2} \\ & = 1 & & = 6 \\ \\ V &= \Big\{ 1 ; 6 \Big\} & &\end{align} \\ -----------------\)
  4. \((-2x+1)(-x+1)-x(x+5)=-8\\ \Leftrightarrow 2x^2-2x-x+1 -x^2-5x+8=0 \\ \Leftrightarrow x^2-6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.9 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.1} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
  5. \(x(x-7)=3(x-3) \\ \Leftrightarrow x^2-7x=3x-9 \\ \Leftrightarrow x^2-10x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.9 & &\\ & = 100-36 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-10)-\sqrt64}{2.1} & & = \frac{-(-10)+\sqrt64}{2.1} \\ & = \frac{2}{2} & & = \frac{18}{2} \\ & = 1 & & = 9 \\ \\ V &= \Big\{ 1 ; 9 \Big\} & &\end{align} \\ -----------------\)
  6. \(-(8-63x)=-9x^2-(89-17x) \\ \Leftrightarrow -8+63x=-9x^2-89+17x \\ \Leftrightarrow 9x^2+46x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+46x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (46)^2-4.9.81 & &\\ & = 2116-2916 & & \\ & = -800 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  7. \(-\frac{5}{4}x=-\frac{1}{4}x^2-1 \\ \Leftrightarrow \frac{1}{4}x^2-\frac{5}{4}x+1=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{5}{4}x+1\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-5x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.4 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt9}{2.1} & & = \frac{-(-5)+\sqrt9}{2.1} \\ & = \frac{2}{2} & & = \frac{8}{2} \\ & = 1 & & = 4 \\ \\ V &= \Big\{ 1 ; 4 \Big\} & &\end{align} \\ -----------------\)
  8. \(x(x+11)=12(x+1) \\ \Leftrightarrow x^2+11x=12x+12 \\ \Leftrightarrow x^2-x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-12) & &\\ & = 1+48 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt49}{2.1} & & = \frac{-(-1)+\sqrt49}{2.1} \\ & = \frac{-6}{2} & & = \frac{8}{2} \\ & = -3 & & = 4 \\ \\ V &= \Big\{ -3 ; 4 \Big\} & &\end{align} \\ -----------------\)
  9. \(x(3x+29)=4(x-12) \\ \Leftrightarrow 3x^2+29x=4x-48 \\ \Leftrightarrow 3x^2+25x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+25x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.3.48 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.3} & & = \frac{-25+\sqrt49}{2.3} \\ & = \frac{-32}{6} & & = \frac{-18}{6} \\ & = \frac{-16}{3} & & = -3 \\ \\ V &= \Big\{ \frac{-16}{3} ; -3 \Big\} & &\end{align} \\ -----------------\)
  10. \(\frac{13}{3}x=-\frac{4}{3}x^2-3 \\ \Leftrightarrow \frac{4}{3}x^2+\frac{13}{3}x+3=0 \\ \Leftrightarrow \color{red}{3.} \left(\frac{4}{3}x^2+\frac{13}{3}x+3\right)=0 \color{red}{.3} \\ \Leftrightarrow 4x^2+13x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+13x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.4.9 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.4} & & = \frac{-13+\sqrt25}{2.4} \\ & = \frac{-18}{8} & & = \frac{-8}{8} \\ & = \frac{-9}{4} & & = -1 \\ \\ V &= \Big\{ \frac{-9}{4} ; -1 \Big\} & &\end{align} \\ -----------------\)
  11. \((-5x-2)(-x-5)-x(-31x+10)=6\\ \Leftrightarrow 5x^2+25x+2x+10 +31x^2-10x-6=0 \\ \Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  12. \((5x+4)(3x-2)-x(14x-21)=2\\ \Leftrightarrow 15x^2-10x+12x-8 -14x^2+21x-2=0 \\ \Leftrightarrow x^2+3x-10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-10) & &\\ & = 9+40 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt49}{2.1} & & = \frac{-3+\sqrt49}{2.1} \\ & = \frac{-10}{2} & & = \frac{4}{2} \\ & = -5 & & = 2 \\ \\ V &= \Big\{ -5 ; 2 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-09-22 07:28:24
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