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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-(2-31x)=-x^2-(34-19x)\)
  2. \(\frac{13}{24}x=-\frac{1}{4}x^2-\frac{1}{4}\)
  3. \(\frac{1}{18}x^2+x+\frac{9}{2}=0\)
  4. \(-\frac{1}{3}x=-\frac{1}{3}x^2-\frac{1}{12}\)
  5. \(x(x+19)=3(x-16)\)
  6. \((3x-3)(-2x-4)-x(-7x+1)=144\)
  7. \((x+4)(4x+5)-x(0x+23)=4\)
  8. \((-x-2)(-2x-3)-x(0x+2)=78\)
  9. \(-x=-\frac{1}{2}x^2+\frac{3}{2}\)
  10. \(\frac{1}{4}x^2+\frac{21}{4}x+\frac{55}{2}=0\)
  11. \((x+1)(-x+5)-x(-3x+7)=7\)
  12. \(\frac{1}{8}x^2+\frac{1}{2}x-4=0\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-(2-31x)=-x^2-(34-19x) \\ \Leftrightarrow -2+31x=-x^2-34+19x \\ \Leftrightarrow x^2+12x+32=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+32=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.32 & &\\ & = 144-128 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-12-\sqrt16}{2.1} & & = \frac{-12+\sqrt16}{2.1} \\ & = \frac{-16}{2} & & = \frac{-8}{2} \\ & = -8 & & = -4 \\ \\ V &= \Big\{ -8 ; -4 \Big\} & &\end{align} \\ -----------------\)
  2. \(\frac{13}{24}x=-\frac{1}{4}x^2-\frac{1}{4} \\ \Leftrightarrow \frac{1}{4}x^2+\frac{13}{24}x+\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{24.} \left(\frac{1}{4}x^2+\frac{13}{24}x+\frac{1}{4}\right)=0 \color{red}{.24} \\ \Leftrightarrow 6x^2+13x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+13x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.6.6 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.6} & & = \frac{-13+\sqrt25}{2.6} \\ & = \frac{-18}{12} & & = \frac{-8}{12} \\ & = \frac{-3}{2} & & = \frac{-2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{-2}{3} \Big\} & &\end{align} \\ -----------------\)
  3. \(\frac{1}{18}x^2+x+\frac{9}{2}=0\\ \Leftrightarrow \color{red}{18.} \left(\frac{1}{18}x^2+x+\frac{9}{2}\right)=0 \color{red}{.18} \\ \Leftrightarrow x^2+18x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+18x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (18)^2-4.1.81 & &\\ & = 324-324 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-18}{2.1} & & \\ & = -9 & & \\V &= \Big\{ -9 \Big\} & &\end{align} \\ -----------------\)
  4. \(-\frac{1}{3}x=-\frac{1}{3}x^2-\frac{1}{12} \\ \Leftrightarrow \frac{1}{3}x^2-\frac{1}{3}x+\frac{1}{12}=0 \\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{3}x^2-\frac{1}{3}x+\frac{1}{12}\right)=0 \color{red}{.12} \\ \Leftrightarrow 16x^2-16x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-16x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.16.4 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.16} & & \\ & = \frac{1}{2} & & \\V &= \Big\{ \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  5. \(x(x+19)=3(x-16) \\ \Leftrightarrow x^2+19x=3x-48 \\ \Leftrightarrow x^2+16x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.48 & &\\ & = 256-192 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-16-\sqrt64}{2.1} & & = \frac{-16+\sqrt64}{2.1} \\ & = \frac{-24}{2} & & = \frac{-8}{2} \\ & = -12 & & = -4 \\ \\ V &= \Big\{ -12 ; -4 \Big\} & &\end{align} \\ -----------------\)
  6. \((3x-3)(-2x-4)-x(-7x+1)=144\\ \Leftrightarrow -6x^2-12x+6x+12 +7x^2-x-144=0 \\ \Leftrightarrow x^2-x-132=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-132=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-132) & &\\ & = 1+528 & & \\ & = 529 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt529}{2.1} & & = \frac{-(-1)+\sqrt529}{2.1} \\ & = \frac{-22}{2} & & = \frac{24}{2} \\ & = -11 & & = 12 \\ \\ V &= \Big\{ -11 ; 12 \Big\} & &\end{align} \\ -----------------\)
  7. \((x+4)(4x+5)-x(0x+23)=4\\ \Leftrightarrow 4x^2+5x+16x+20 +0x^2-23x-4=0 \\ \Leftrightarrow 4x^2+2x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+2x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.4.16 & &\\ & = 4-256 & & \\ & = -252 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  8. \((-x-2)(-2x-3)-x(0x+2)=78\\ \Leftrightarrow 2x^2+3x+4x+6 +0x^2-2x-78=0 \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
  9. \(-x=-\frac{1}{2}x^2+\frac{3}{2} \\ \Leftrightarrow \frac{1}{2}x^2-x-\frac{3}{2}=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-x-\frac{3}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2-2x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-3) & &\\ & = 4+12 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt16}{2.1} & & = \frac{-(-2)+\sqrt16}{2.1} \\ & = \frac{-2}{2} & & = \frac{6}{2} \\ & = -1 & & = 3 \\ \\ V &= \Big\{ -1 ; 3 \Big\} & &\end{align} \\ -----------------\)
  10. \(\frac{1}{4}x^2+\frac{21}{4}x+\frac{55}{2}=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{21}{4}x+\frac{55}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+21x+110=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+21x+110=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (21)^2-4.1.110 & &\\ & = 441-440 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-21-\sqrt1}{2.1} & & = \frac{-21+\sqrt1}{2.1} \\ & = \frac{-22}{2} & & = \frac{-20}{2} \\ & = -11 & & = -10 \\ \\ V &= \Big\{ -11 ; -10 \Big\} & &\end{align} \\ -----------------\)
  11. \((x+1)(-x+5)-x(-3x+7)=7\\ \Leftrightarrow -x^2+5x-x+5 +3x^2-7x-7=0 \\ \Leftrightarrow 2x^2+3x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+3x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.2.(-2) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.2} & & = \frac{-3+\sqrt25}{2.2} \\ & = \frac{-8}{4} & & = \frac{2}{4} \\ & = -2 & & = \frac{1}{2} \\ \\ V &= \Big\{ -2 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  12. \(\frac{1}{8}x^2+\frac{1}{2}x-4=0\\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2+\frac{1}{2}x-4\right)=0 \color{red}{.8} \\ \Leftrightarrow x^2+4x-32=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-32=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-32) & &\\ & = 16+128 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt144}{2.1} & & = \frac{-4+\sqrt144}{2.1} \\ & = \frac{-16}{2} & & = \frac{8}{2} \\ & = -8 & & = 4 \\ \\ V &= \Big\{ -8 ; 4 \Big\} & &\end{align} \\ -----------------\)
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