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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(10x^2-(4x-100)=x(x+18)\)
  2. \(\frac{23}{8}x=-x^2-\frac{25}{4}\)
  3. \(\frac{1}{4}x=-\frac{1}{4}x^2+18\)
  4. \((-4x+1)(-4x-4)-x(15x+34)=-125\)
  5. \(x^2+\frac{3}{4}x-\frac{1}{4}=0\)
  6. \(\frac{3}{2}x=-\frac{1}{2}x^2+5\)
  7. \(\frac{4}{3}x=-\frac{1}{9}x^2-3\)
  8. \((2x+3)(3x+2)-x(-10x+98)=-115\)
  9. \(x(4x+43)=36(x+1)\)
  10. \(x(9x-55)=11(x-11)\)
  11. \(-\frac{1}{2}x=-\frac{1}{16}x^2-1\)
  12. \((-5x+2)(x+3)-x(-6x-13)=2\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(10x^2-(4x-100)=x(x+18) \\ \Leftrightarrow 10x^2-4x+100=x^2+18x \\ \Leftrightarrow 9x^2-22x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-22x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.9.100 & &\\ & = 484-3600 & & \\ & = -3116 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  2. \(\frac{23}{8}x=-x^2-\frac{25}{4} \\ \Leftrightarrow x^2+\frac{23}{8}x+\frac{25}{4}=0 \\ \Leftrightarrow \color{red}{8.} \left(x^2+\frac{23}{8}x+\frac{25}{4}\right)=0 \color{red}{.8} \\ \Leftrightarrow 16x^2+46x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+46x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (46)^2-4.16.100 & &\\ & = 2116-6400 & & \\ & = -4284 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  3. \(\frac{1}{4}x=-\frac{1}{4}x^2+18 \\ \Leftrightarrow \frac{1}{4}x^2+\frac{1}{4}x-18=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{1}{4}x-18\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-72) & &\\ & = 1+288 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt289}{2.1} & & = \frac{-1+\sqrt289}{2.1} \\ & = \frac{-18}{2} & & = \frac{16}{2} \\ & = -9 & & = 8 \\ \\ V &= \Big\{ -9 ; 8 \Big\} & &\end{align} \\ -----------------\)
  4. \((-4x+1)(-4x-4)-x(15x+34)=-125\\ \Leftrightarrow 16x^2+16x-4x-4 -15x^2-34x+125=0 \\ \Leftrightarrow x^2-22x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-22x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.1.121 & &\\ & = 484-484 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-22)}{2.1} & & \\ & = 11 & & \\V &= \Big\{ 11 \Big\} & &\end{align} \\ -----------------\)
  5. \(x^2+\frac{3}{4}x-\frac{1}{4}=0\\ \Leftrightarrow \color{red}{4.} \left(x^2+\frac{3}{4}x-\frac{1}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow 4x^2+3x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+3x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.4.(-1) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.4} & & = \frac{-3+\sqrt25}{2.4} \\ & = \frac{-8}{8} & & = \frac{2}{8} \\ & = -1 & & = \frac{1}{4} \\ \\ V &= \Big\{ -1 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
  6. \(\frac{3}{2}x=-\frac{1}{2}x^2+5 \\ \Leftrightarrow \frac{1}{2}x^2+\frac{3}{2}x-5=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+\frac{3}{2}x-5\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2+3x-10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-10) & &\\ & = 9+40 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt49}{2.1} & & = \frac{-3+\sqrt49}{2.1} \\ & = \frac{-10}{2} & & = \frac{4}{2} \\ & = -5 & & = 2 \\ \\ V &= \Big\{ -5 ; 2 \Big\} & &\end{align} \\ -----------------\)
  7. \(\frac{4}{3}x=-\frac{1}{9}x^2-3 \\ \Leftrightarrow \frac{1}{9}x^2+\frac{4}{3}x+3=0 \\ \Leftrightarrow \color{red}{9.} \left(\frac{1}{9}x^2+\frac{4}{3}x+3\right)=0 \color{red}{.9} \\ \Leftrightarrow x^2+12x+27=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+27=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.27 & &\\ & = 144-108 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-12-\sqrt36}{2.1} & & = \frac{-12+\sqrt36}{2.1} \\ & = \frac{-18}{2} & & = \frac{-6}{2} \\ & = -9 & & = -3 \\ \\ V &= \Big\{ -9 ; -3 \Big\} & &\end{align} \\ -----------------\)
  8. \((2x+3)(3x+2)-x(-10x+98)=-115\\ \Leftrightarrow 6x^2+4x+9x+6 +10x^2-98x+115=0 \\ \Leftrightarrow 16x^2-88x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-88x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-88)^2-4.16.121 & &\\ & = 7744-7744 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-88)}{2.16} & & \\ & = \frac{11}{4} & & \\V &= \Big\{ \frac{11}{4} \Big\} & &\end{align} \\ -----------------\)
  9. \(x(4x+43)=36(x+1) \\ \Leftrightarrow 4x^2+43x=36x+36 \\ \Leftrightarrow 4x^2+7x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+7x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.4.(-36) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.4} & & = \frac{-7+\sqrt625}{2.4} \\ & = \frac{-32}{8} & & = \frac{18}{8} \\ & = -4 & & = \frac{9}{4} \\ \\ V &= \Big\{ -4 ; \frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
  10. \(x(9x-55)=11(x-11) \\ \Leftrightarrow 9x^2-55x=11x-121 \\ \Leftrightarrow 9x^2-66x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-66x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-66)^2-4.9.121 & &\\ & = 4356-4356 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-66)}{2.9} & & \\ & = \frac{11}{3} & & \\V &= \Big\{ \frac{11}{3} \Big\} & &\end{align} \\ -----------------\)
  11. \(-\frac{1}{2}x=-\frac{1}{16}x^2-1 \\ \Leftrightarrow \frac{1}{16}x^2-\frac{1}{2}x+1=0 \\ \Leftrightarrow \color{red}{16.} \left(\frac{1}{16}x^2-\frac{1}{2}x+1\right)=0 \color{red}{.16} \\ \Leftrightarrow 4x^2-32x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-32x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-32)^2-4.4.64 & &\\ & = 1024-1024 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-32)}{2.4} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
  12. \((-5x+2)(x+3)-x(-6x-13)=2\\ \Leftrightarrow -5x^2-15x+2x+6 +6x^2+13x-2=0 \\ \Leftrightarrow x^2+4x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.1} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
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