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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-2x=-\frac{1}{2}x^2+\frac{77}{2}\)
  2. \(-(7-33x)=-9x^2-(11-20x)\)
  3. \(-(5-30x)=-12x^2-(8-17x)\)
  4. \(\frac{3}{5}x=-\frac{2}{5}x^2-\frac{9}{40}\)
  5. \(-(2+30x)=-9x^2-(38-6x)\)
  6. \(x(4x+41)=9(x-9)\)
  7. \(-(7-10x)=-x^2-(-41-2x)\)
  8. \(\frac{17}{5}x=-\frac{1}{5}x^2-\frac{66}{5}\)
  9. \(\frac{1}{24}x^2-\frac{1}{4}x+\frac{3}{2}=0\)
  10. \(\frac{4}{15}x^2-\frac{4}{3}x+\frac{5}{3}=0\)
  11. \(\frac{16}{3}x=-\frac{1}{3}x^2-21\)
  12. \(\frac{1}{16}x^2-\frac{1}{4}x-2=0\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-2x=-\frac{1}{2}x^2+\frac{77}{2} \\ \Leftrightarrow \frac{1}{2}x^2-2x-\frac{77}{2}=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-2x-\frac{77}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2-4x-77=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-77=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-77) & &\\ & = 16+308 & & \\ & = 324 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt324}{2.1} & & = \frac{-(-4)+\sqrt324}{2.1} \\ & = \frac{-14}{2} & & = \frac{22}{2} \\ & = -7 & & = 11 \\ \\ V &= \Big\{ -7 ; 11 \Big\} & &\end{align} \\ -----------------\)
  2. \(-(7-33x)=-9x^2-(11-20x) \\ \Leftrightarrow -7+33x=-9x^2-11+20x \\ \Leftrightarrow 9x^2+13x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+13x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.9.4 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.9} & & = \frac{-13+\sqrt25}{2.9} \\ & = \frac{-18}{18} & & = \frac{-8}{18} \\ & = -1 & & = \frac{-4}{9} \\ \\ V &= \Big\{ -1 ; \frac{-4}{9} \Big\} & &\end{align} \\ -----------------\)
  3. \(-(5-30x)=-12x^2-(8-17x) \\ \Leftrightarrow -5+30x=-12x^2-8+17x \\ \Leftrightarrow 12x^2+13x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+13x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.12.3 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.12} & & = \frac{-13+\sqrt25}{2.12} \\ & = \frac{-18}{24} & & = \frac{-8}{24} \\ & = \frac{-3}{4} & & = \frac{-1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{-1}{3} \Big\} & &\end{align} \\ -----------------\)
  4. \(\frac{3}{5}x=-\frac{2}{5}x^2-\frac{9}{40} \\ \Leftrightarrow \frac{2}{5}x^2+\frac{3}{5}x+\frac{9}{40}=0 \\ \Leftrightarrow \color{red}{40.} \left(\frac{2}{5}x^2+\frac{3}{5}x+\frac{9}{40}\right)=0 \color{red}{.40} \\ \Leftrightarrow 16x^2+24x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+24x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.16.9 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.16} & & \\ & = -\frac{3}{4} & & \\V &= \Big\{ -\frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
  5. \(-(2+30x)=-9x^2-(38-6x) \\ \Leftrightarrow -2-30x=-9x^2-38+6x \\ \Leftrightarrow 9x^2-36x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-36x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-36)^2-4.9.36 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-36)}{2.9} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
  6. \(x(4x+41)=9(x-9) \\ \Leftrightarrow 4x^2+41x=9x-81 \\ \Leftrightarrow 4x^2+32x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+32x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (32)^2-4.4.81 & &\\ & = 1024-1296 & & \\ & = -272 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  7. \(-(7-10x)=-x^2-(-41-2x) \\ \Leftrightarrow -7+10x=-x^2+41+2x \\ \Leftrightarrow x^2+8x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.(-48) & &\\ & = 64+192 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-8-\sqrt256}{2.1} & & = \frac{-8+\sqrt256}{2.1} \\ & = \frac{-24}{2} & & = \frac{8}{2} \\ & = -12 & & = 4 \\ \\ V &= \Big\{ -12 ; 4 \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{17}{5}x=-\frac{1}{5}x^2-\frac{66}{5} \\ \Leftrightarrow \frac{1}{5}x^2+\frac{17}{5}x+\frac{66}{5}=0 \\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+\frac{17}{5}x+\frac{66}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2+17x+66=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+17x+66=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.1.66 & &\\ & = 289-264 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt25}{2.1} & & = \frac{-17+\sqrt25}{2.1} \\ & = \frac{-22}{2} & & = \frac{-12}{2} \\ & = -11 & & = -6 \\ \\ V &= \Big\{ -11 ; -6 \Big\} & &\end{align} \\ -----------------\)
  9. \(\frac{1}{24}x^2-\frac{1}{4}x+\frac{3}{2}=0\\ \Leftrightarrow \color{red}{24.} \left(\frac{1}{24}x^2-\frac{1}{4}x+\frac{3}{2}\right)=0 \color{red}{.24} \\ \Leftrightarrow x^2-6x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.36 & &\\ & = 36-144 & & \\ & = -108 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  10. \(\frac{4}{15}x^2-\frac{4}{3}x+\frac{5}{3}=0\\ \Leftrightarrow \color{red}{15.} \left(\frac{4}{15}x^2-\frac{4}{3}x+\frac{5}{3}\right)=0 \color{red}{.15} \\ \Leftrightarrow 16x^2-80x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-80x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-80)^2-4.16.100 & &\\ & = 6400-6400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-80)}{2.16} & & \\ & = \frac{5}{2} & & \\V &= \Big\{ \frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
  11. \(\frac{16}{3}x=-\frac{1}{3}x^2-21 \\ \Leftrightarrow \frac{1}{3}x^2+\frac{16}{3}x+21=0 \\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+\frac{16}{3}x+21\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2+16x+63=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+63=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.63 & &\\ & = 256-252 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-16-\sqrt4}{2.1} & & = \frac{-16+\sqrt4}{2.1} \\ & = \frac{-18}{2} & & = \frac{-14}{2} \\ & = -9 & & = -7 \\ \\ V &= \Big\{ -9 ; -7 \Big\} & &\end{align} \\ -----------------\)
  12. \(\frac{1}{16}x^2-\frac{1}{4}x-2=0\\ \Leftrightarrow \color{red}{16.} \left(\frac{1}{16}x^2-\frac{1}{4}x-2\right)=0 \color{red}{.16} \\ \Leftrightarrow x^2-4x-32=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-32=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-32) & &\\ & = 16+128 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt144}{2.1} & & = \frac{-(-4)+\sqrt144}{2.1} \\ & = \frac{-8}{2} & & = \frac{16}{2} \\ & = -4 & & = 8 \\ \\ V &= \Big\{ -4 ; 8 \Big\} & &\end{align} \\ -----------------\)
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