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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(7x^2-(17x-49)=3x(x-15)\)
  2. \(\frac{25}{24}x=-\frac{1}{4}x^2-1\)
  3. \(-\frac{3}{4}x=-\frac{1}{4}x^2+22\)
  4. \(-(3-16x)=-x^2-(-21-11x)\)
  5. \(-(14-23x)=-x^2-(-63-19x)\)
  6. \(-(3-19x)=-x^2-(30-7x)\)
  7. \(\frac{1}{4}x^2+\frac{1}{2}x-\frac{3}{4}=0\)
  8. \(\frac{7}{6}x=-\frac{1}{2}x^2+8\)
  9. \(33x^2-(12x-49)=17x(x-4)\)
  10. \(x(x+1)=-5(x+1)\)
  11. \((-2x+5)(2x-5)-x(-5x-5)=-46\)
  12. \(x(x-6)=4(x-6)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(7x^2-(17x-49)=3x(x-15) \\ \Leftrightarrow 7x^2-17x+49=3x^2-45x \\ \Leftrightarrow 4x^2+28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+28x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (28)^2-4.4.49 & &\\ & = 784-784 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-28}{2.4} & & \\ & = -\frac{7}{2} & & \\V &= \Big\{ -\frac{7}{2} \Big\} & &\end{align} \\ -----------------\)
  2. \(\frac{25}{24}x=-\frac{1}{4}x^2-1 \\ \Leftrightarrow \frac{1}{4}x^2+\frac{25}{24}x+1=0 \\ \Leftrightarrow \color{red}{24.} \left(\frac{1}{4}x^2+\frac{25}{24}x+1\right)=0 \color{red}{.24} \\ \Leftrightarrow 6x^2+25x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+25x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.6.24 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.6} & & = \frac{-25+\sqrt49}{2.6} \\ & = \frac{-32}{12} & & = \frac{-18}{12} \\ & = \frac{-8}{3} & & = \frac{-3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{-3}{2} \Big\} & &\end{align} \\ -----------------\)
  3. \(-\frac{3}{4}x=-\frac{1}{4}x^2+22 \\ \Leftrightarrow \frac{1}{4}x^2-\frac{3}{4}x-22=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{3}{4}x-22\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-3x-88=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-88=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-88) & &\\ & = 9+352 & & \\ & = 361 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt361}{2.1} & & = \frac{-(-3)+\sqrt361}{2.1} \\ & = \frac{-16}{2} & & = \frac{22}{2} \\ & = -8 & & = 11 \\ \\ V &= \Big\{ -8 ; 11 \Big\} & &\end{align} \\ -----------------\)
  4. \(-(3-16x)=-x^2-(-21-11x) \\ \Leftrightarrow -3+16x=-x^2+21+11x \\ \Leftrightarrow x^2+5x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-24) & &\\ & = 25+96 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt121}{2.1} & & = \frac{-5+\sqrt121}{2.1} \\ & = \frac{-16}{2} & & = \frac{6}{2} \\ & = -8 & & = 3 \\ \\ V &= \Big\{ -8 ; 3 \Big\} & &\end{align} \\ -----------------\)
  5. \(-(14-23x)=-x^2-(-63-19x) \\ \Leftrightarrow -14+23x=-x^2+63+19x \\ \Leftrightarrow x^2+4x-77=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-77=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-77) & &\\ & = 16+308 & & \\ & = 324 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt324}{2.1} & & = \frac{-4+\sqrt324}{2.1} \\ & = \frac{-22}{2} & & = \frac{14}{2} \\ & = -11 & & = 7 \\ \\ V &= \Big\{ -11 ; 7 \Big\} & &\end{align} \\ -----------------\)
  6. \(-(3-19x)=-x^2-(30-7x) \\ \Leftrightarrow -3+19x=-x^2-30+7x \\ \Leftrightarrow x^2+12x+27=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+27=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.27 & &\\ & = 144-108 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-12-\sqrt36}{2.1} & & = \frac{-12+\sqrt36}{2.1} \\ & = \frac{-18}{2} & & = \frac{-6}{2} \\ & = -9 & & = -3 \\ \\ V &= \Big\{ -9 ; -3 \Big\} & &\end{align} \\ -----------------\)
  7. \(\frac{1}{4}x^2+\frac{1}{2}x-\frac{3}{4}=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{1}{2}x-\frac{3}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+2x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-3) & &\\ & = 4+12 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt16}{2.1} & & = \frac{-2+\sqrt16}{2.1} \\ & = \frac{-6}{2} & & = \frac{2}{2} \\ & = -3 & & = 1 \\ \\ V &= \Big\{ -3 ; 1 \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{7}{6}x=-\frac{1}{2}x^2+8 \\ \Leftrightarrow \frac{1}{2}x^2+\frac{7}{6}x-8=0 \\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{2}x^2+\frac{7}{6}x-8\right)=0 \color{red}{.6} \\ \Leftrightarrow 3x^2+7x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+7x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.3.(-48) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.3} & & = \frac{-7+\sqrt625}{2.3} \\ & = \frac{-32}{6} & & = \frac{18}{6} \\ & = \frac{-16}{3} & & = 3 \\ \\ V &= \Big\{ \frac{-16}{3} ; 3 \Big\} & &\end{align} \\ -----------------\)
  9. \(33x^2-(12x-49)=17x(x-4) \\ \Leftrightarrow 33x^2-12x+49=17x^2-68x \\ \Leftrightarrow 16x^2+56x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+56x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (56)^2-4.16.49 & &\\ & = 3136-3136 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-56}{2.16} & & \\ & = -\frac{7}{4} & & \\V &= \Big\{ -\frac{7}{4} \Big\} & &\end{align} \\ -----------------\)
  10. \(x(x+1)=-5(x+1) \\ \Leftrightarrow x^2+x=-5x-5 \\ \Leftrightarrow x^2+6x+5=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+5=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.5 & &\\ & = 36-20 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt16}{2.1} & & = \frac{-6+\sqrt16}{2.1} \\ & = \frac{-10}{2} & & = \frac{-2}{2} \\ & = -5 & & = -1 \\ \\ V &= \Big\{ -5 ; -1 \Big\} & &\end{align} \\ -----------------\)
  11. \((-2x+5)(2x-5)-x(-5x-5)=-46\\ \Leftrightarrow -4x^2+10x+10x-25 +5x^2+5x+46=0 \\ \Leftrightarrow x^2-10x+21=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x+21=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.21 & &\\ & = 100-84 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-10)-\sqrt16}{2.1} & & = \frac{-(-10)+\sqrt16}{2.1} \\ & = \frac{6}{2} & & = \frac{14}{2} \\ & = 3 & & = 7 \\ \\ V &= \Big\{ 3 ; 7 \Big\} & &\end{align} \\ -----------------\)
  12. \(x(x-6)=4(x-6) \\ \Leftrightarrow x^2-6x=4x-24 \\ \Leftrightarrow x^2-10x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.24 & &\\ & = 100-96 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-10)-\sqrt4}{2.1} & & = \frac{-(-10)+\sqrt4}{2.1} \\ & = \frac{8}{2} & & = \frac{12}{2} \\ & = 4 & & = 6 \\ \\ V &= \Big\{ 4 ; 6 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2025-11-15 20:30:35
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