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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{11}{4}x=-\frac{1}{4}x^2-\frac{5}{2}\)
  2. \(-\frac{7}{5}x=-\frac{1}{10}x^2-\frac{24}{5}\)
  3. \(3x^2-(9x+8)=x(x-24)\)
  4. \(x(x+49)=45(x+1)\)
  5. \(x(9x-7)=-(x+1)\)
  6. \(-(15-24x)=-x^2-(40-18x)\)
  7. \(x(16x+63)=7(x-7)\)
  8. \(\frac{1}{60}x^2+\frac{1}{5}x+\frac{3}{5}=0\)
  9. \(x(3x+55)=48(x+1)\)
  10. \(-(4-54x)=-4x^2-(125-10x)\)
  11. \((-x+3)(-2x-5)-x(-2x-38)=-64\)
  12. \((-4x-1)(x-1)-x(-28x-2)=7\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{11}{4}x=-\frac{1}{4}x^2-\frac{5}{2} \\ \Leftrightarrow \frac{1}{4}x^2+\frac{11}{4}x+\frac{5}{2}=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{11}{4}x+\frac{5}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+11x+10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.10 & &\\ & = 121-40 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt81}{2.1} & & = \frac{-11+\sqrt81}{2.1} \\ & = \frac{-20}{2} & & = \frac{-2}{2} \\ & = -10 & & = -1 \\ \\ V &= \Big\{ -10 ; -1 \Big\} & &\end{align} \\ -----------------\)
  2. \(-\frac{7}{5}x=-\frac{1}{10}x^2-\frac{24}{5} \\ \Leftrightarrow \frac{1}{10}x^2-\frac{7}{5}x+\frac{24}{5}=0 \\ \Leftrightarrow \color{red}{10.} \left(\frac{1}{10}x^2-\frac{7}{5}x+\frac{24}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow x^2-14x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.48 & &\\ & = 196-192 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-14)-\sqrt4}{2.1} & & = \frac{-(-14)+\sqrt4}{2.1} \\ & = \frac{12}{2} & & = \frac{16}{2} \\ & = 6 & & = 8 \\ \\ V &= \Big\{ 6 ; 8 \Big\} & &\end{align} \\ -----------------\)
  3. \(3x^2-(9x+8)=x(x-24) \\ \Leftrightarrow 3x^2-9x-8=x^2-24x \\ \Leftrightarrow 2x^2+15x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+15x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.2.(-8) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.2} & & = \frac{-15+\sqrt289}{2.2} \\ & = \frac{-32}{4} & & = \frac{2}{4} \\ & = -8 & & = \frac{1}{2} \\ \\ V &= \Big\{ -8 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  4. \(x(x+49)=45(x+1) \\ \Leftrightarrow x^2+49x=45x+45 \\ \Leftrightarrow x^2+4x-45=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-45=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-45) & &\\ & = 16+180 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt196}{2.1} & & = \frac{-4+\sqrt196}{2.1} \\ & = \frac{-18}{2} & & = \frac{10}{2} \\ & = -9 & & = 5 \\ \\ V &= \Big\{ -9 ; 5 \Big\} & &\end{align} \\ -----------------\)
  5. \(x(9x-7)=-(x+1) \\ \Leftrightarrow 9x^2-7x=-x-1 \\ \Leftrightarrow 9x^2-6x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-6x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.9.1 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.9} & & \\ & = \frac{1}{3} & & \\V &= \Big\{ \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
  6. \(-(15-24x)=-x^2-(40-18x) \\ \Leftrightarrow -15+24x=-x^2-40+18x \\ \Leftrightarrow x^2+6x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.25 & &\\ & = 36-100 & & \\ & = -64 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  7. \(x(16x+63)=7(x-7) \\ \Leftrightarrow 16x^2+63x=7x-49 \\ \Leftrightarrow 16x^2+56x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+56x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (56)^2-4.16.49 & &\\ & = 3136-3136 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-56}{2.16} & & \\ & = -\frac{7}{4} & & \\V &= \Big\{ -\frac{7}{4} \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{1}{60}x^2+\frac{1}{5}x+\frac{3}{5}=0\\ \Leftrightarrow \color{red}{60.} \left(\frac{1}{60}x^2+\frac{1}{5}x+\frac{3}{5}\right)=0 \color{red}{.60} \\ \Leftrightarrow 4x^2+48x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+48x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (48)^2-4.4.144 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-48}{2.4} & & \\ & = -6 & & \\V &= \Big\{ -6 \Big\} & &\end{align} \\ -----------------\)
  9. \(x(3x+55)=48(x+1) \\ \Leftrightarrow 3x^2+55x=48x+48 \\ \Leftrightarrow 3x^2+7x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+7x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.3.(-48) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.3} & & = \frac{-7+\sqrt625}{2.3} \\ & = \frac{-32}{6} & & = \frac{18}{6} \\ & = \frac{-16}{3} & & = 3 \\ \\ V &= \Big\{ \frac{-16}{3} ; 3 \Big\} & &\end{align} \\ -----------------\)
  10. \(-(4-54x)=-4x^2-(125-10x) \\ \Leftrightarrow -4+54x=-4x^2-125+10x \\ \Leftrightarrow 4x^2+44x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+44x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (44)^2-4.4.121 & &\\ & = 1936-1936 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-44}{2.4} & & \\ & = -\frac{11}{2} & & \\V &= \Big\{ -\frac{11}{2} \Big\} & &\end{align} \\ -----------------\)
  11. \((-x+3)(-2x-5)-x(-2x-38)=-64\\ \Leftrightarrow 2x^2+5x-6x-15 +2x^2+38x+64=0 \\ \Leftrightarrow 4x^2+28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+28x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (28)^2-4.4.49 & &\\ & = 784-784 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-28}{2.4} & & \\ & = -\frac{7}{2} & & \\V &= \Big\{ -\frac{7}{2} \Big\} & &\end{align} \\ -----------------\)
  12. \((-4x-1)(x-1)-x(-28x-2)=7\\ \Leftrightarrow -4x^2+4x-x+1 +28x^2+2x-7=0 \\ \Leftrightarrow 24x^2+7x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+7x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.24.(-6) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.24} & & = \frac{-7+\sqrt625}{2.24} \\ & = \frac{-32}{48} & & = \frac{18}{48} \\ & = \frac{-2}{3} & & = \frac{3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{3}{8} \Big\} & &\end{align} \\ -----------------\)
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