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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-(14-22x)=-9x^2-(18-9x)\)
  2. \(x(4x+14)=9(x+1)\)
  3. \(-(5+9x)=-x^2-(77-8x)\)
  4. \((x-1)(-5x-2)-x(-21x+34)=-98\)
  5. \((-4x-4)(4x+3)-x(-17x-20)=48\)
  6. \(\frac{1}{3}x^2-x+\frac{3}{4}=0\)
  7. \(5x^2-(13x-36)=x(x+11)\)
  8. \(\frac{1}{2}x^2-\frac{7}{2}x-30=0\)
  9. \(-(7-3x)=-x^2-(-9-9x)\)
  10. \(\frac{15}{4}x=-\frac{1}{4}x^2-11\)
  11. \(-\frac{1}{2}x=-\frac{1}{8}x^2-\frac{1}{2}\)
  12. \(-(15-3x)=-9x^2-(16-9x)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-(14-22x)=-9x^2-(18-9x) \\ \Leftrightarrow -14+22x=-9x^2-18+9x \\ \Leftrightarrow 9x^2+13x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+13x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.9.4 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.9} & & = \frac{-13+\sqrt25}{2.9} \\ & = \frac{-18}{18} & & = \frac{-8}{18} \\ & = -1 & & = \frac{-4}{9} \\ \\ V &= \Big\{ -1 ; \frac{-4}{9} \Big\} & &\end{align} \\ -----------------\)
  2. \(x(4x+14)=9(x+1) \\ \Leftrightarrow 4x^2+14x=9x+9 \\ \Leftrightarrow 4x^2+5x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.(-9) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.4} & & = \frac{-5+\sqrt169}{2.4} \\ & = \frac{-18}{8} & & = \frac{8}{8} \\ & = \frac{-9}{4} & & = 1 \\ \\ V &= \Big\{ \frac{-9}{4} ; 1 \Big\} & &\end{align} \\ -----------------\)
  3. \(-(5+9x)=-x^2-(77-8x) \\ \Leftrightarrow -5-9x=-x^2-77+8x \\ \Leftrightarrow x^2-17x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-17x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-17)^2-4.1.72 & &\\ & = 289-288 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-17)-\sqrt1}{2.1} & & = \frac{-(-17)+\sqrt1}{2.1} \\ & = \frac{16}{2} & & = \frac{18}{2} \\ & = 8 & & = 9 \\ \\ V &= \Big\{ 8 ; 9 \Big\} & &\end{align} \\ -----------------\)
  4. \((x-1)(-5x-2)-x(-21x+34)=-98\\ \Leftrightarrow -5x^2-2x+5x+2 +21x^2-34x+98=0 \\ \Leftrightarrow 16x^2-34x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-34x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-34)^2-4.16.100 & &\\ & = 1156-6400 & & \\ & = -5244 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  5. \((-4x-4)(4x+3)-x(-17x-20)=48\\ \Leftrightarrow -16x^2-12x-16x-12 +17x^2+20x-48=0 \\ \Leftrightarrow x^2-4x-60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-60) & &\\ & = 16+240 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt256}{2.1} & & = \frac{-(-4)+\sqrt256}{2.1} \\ & = \frac{-12}{2} & & = \frac{20}{2} \\ & = -6 & & = 10 \\ \\ V &= \Big\{ -6 ; 10 \Big\} & &\end{align} \\ -----------------\)
  6. \(\frac{1}{3}x^2-x+\frac{3}{4}=0\\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{3}x^2-x+\frac{3}{4}\right)=0 \color{red}{.12} \\ \Leftrightarrow 16x^2-48x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-48x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-48)^2-4.16.36 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-48)}{2.16} & & \\ & = \frac{3}{2} & & \\V &= \Big\{ \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
  7. \(5x^2-(13x-36)=x(x+11) \\ \Leftrightarrow 5x^2-13x+36=x^2+11x \\ \Leftrightarrow 4x^2-24x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-24x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.4.36 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-24)}{2.4} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{1}{2}x^2-\frac{7}{2}x-30=0\\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-\frac{7}{2}x-30\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2-7x-60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x-60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.(-60) & &\\ & = 49+240 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt289}{2.1} & & = \frac{-(-7)+\sqrt289}{2.1} \\ & = \frac{-10}{2} & & = \frac{24}{2} \\ & = -5 & & = 12 \\ \\ V &= \Big\{ -5 ; 12 \Big\} & &\end{align} \\ -----------------\)
  9. \(-(7-3x)=-x^2-(-9-9x) \\ \Leftrightarrow -7+3x=-x^2+9+9x \\ \Leftrightarrow x^2-6x-16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.(-16) & &\\ & = 36+64 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt100}{2.1} & & = \frac{-(-6)+\sqrt100}{2.1} \\ & = \frac{-4}{2} & & = \frac{16}{2} \\ & = -2 & & = 8 \\ \\ V &= \Big\{ -2 ; 8 \Big\} & &\end{align} \\ -----------------\)
  10. \(\frac{15}{4}x=-\frac{1}{4}x^2-11 \\ \Leftrightarrow \frac{1}{4}x^2+\frac{15}{4}x+11=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{15}{4}x+11\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+15x+44=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+15x+44=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.1.44 & &\\ & = 225-176 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt49}{2.1} & & = \frac{-15+\sqrt49}{2.1} \\ & = \frac{-22}{2} & & = \frac{-8}{2} \\ & = -11 & & = -4 \\ \\ V &= \Big\{ -11 ; -4 \Big\} & &\end{align} \\ -----------------\)
  11. \(-\frac{1}{2}x=-\frac{1}{8}x^2-\frac{1}{2} \\ \Leftrightarrow \frac{1}{8}x^2-\frac{1}{2}x+\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2-\frac{1}{2}x+\frac{1}{2}\right)=0 \color{red}{.8} \\ \Leftrightarrow x^2-4x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-4)}{2.1} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
  12. \(-(15-3x)=-9x^2-(16-9x) \\ \Leftrightarrow -15+3x=-9x^2-16+9x \\ \Leftrightarrow 9x^2-6x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-6x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.9.1 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.9} & & \\ & = \frac{1}{3} & & \\V &= \Big\{ \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2025-11-28 15:55:45
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