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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{1}{4}x^2+\frac{3}{8}x-\frac{1}{4}=0\)
  2. \(\frac{5}{6}x=-\frac{4}{5}x^2-\frac{1}{5}\)
  3. \(9x^2-(4x+18)=x(x-11)\)
  4. \(\frac{9}{2}x^2+\frac{25}{4}x+2=0\)
  5. \(\frac{7}{24}x=-\frac{1}{4}x^2+1\)
  6. \(\frac{1}{3}x^2+\frac{4}{3}x-15=0\)
  7. \(-8x=-\frac{1}{2}x^2-\frac{63}{2}\)
  8. \(x(4x+4)=(x+1)\)
  9. \((-5x+3)(-x-4)-x(4x+22)=-52\)
  10. \(\frac{1}{4}x^2+\frac{13}{12}x+1=0\)
  11. \(\frac{1}{48}x^2-\frac{1}{4}x+\frac{3}{4}=0\)
  12. \((-3x+4)(4x+2)-x(-21x+74)=-136\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{1}{4}x^2+\frac{3}{8}x-\frac{1}{4}=0\\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{4}x^2+\frac{3}{8}x-\frac{1}{4}\right)=0 \color{red}{.8} \\ \Leftrightarrow 2x^2+3x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+3x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.2.(-2) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.2} & & = \frac{-3+\sqrt25}{2.2} \\ & = \frac{-8}{4} & & = \frac{2}{4} \\ & = -2 & & = \frac{1}{2} \\ \\ V &= \Big\{ -2 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
  2. \(\frac{5}{6}x=-\frac{4}{5}x^2-\frac{1}{5} \\ \Leftrightarrow \frac{4}{5}x^2+\frac{5}{6}x+\frac{1}{5}=0 \\ \Leftrightarrow \color{red}{30.} \left(\frac{4}{5}x^2+\frac{5}{6}x+\frac{1}{5}\right)=0 \color{red}{.30} \\ \Leftrightarrow 24x^2+25x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+25x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.24.6 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.24} & & = \frac{-25+\sqrt49}{2.24} \\ & = \frac{-32}{48} & & = \frac{-18}{48} \\ & = \frac{-2}{3} & & = \frac{-3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{-3}{8} \Big\} & &\end{align} \\ -----------------\)
  3. \(9x^2-(4x+18)=x(x-11) \\ \Leftrightarrow 9x^2-4x-18=x^2-11x \\ \Leftrightarrow 8x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.8.(-18) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\ & = \frac{-32}{16} & & = \frac{18}{16} \\ & = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)
  4. \(\frac{9}{2}x^2+\frac{25}{4}x+2=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{9}{2}x^2+\frac{25}{4}x+2\right)=0 \color{red}{.4} \\ \Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{7}{24}x=-\frac{1}{4}x^2+1 \\ \Leftrightarrow \frac{1}{4}x^2+\frac{7}{24}x-1=0 \\ \Leftrightarrow \color{red}{24.} \left(\frac{1}{4}x^2+\frac{7}{24}x-1\right)=0 \color{red}{.24} \\ \Leftrightarrow 6x^2+7x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+7x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.6.(-24) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.6} & & = \frac{-7+\sqrt625}{2.6} \\ & = \frac{-32}{12} & & = \frac{18}{12} \\ & = \frac{-8}{3} & & = \frac{3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
  6. \(\frac{1}{3}x^2+\frac{4}{3}x-15=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+\frac{4}{3}x-15\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2+4x-45=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-45=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-45) & &\\ & = 16+180 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt196}{2.1} & & = \frac{-4+\sqrt196}{2.1} \\ & = \frac{-18}{2} & & = \frac{10}{2} \\ & = -9 & & = 5 \\ \\ V &= \Big\{ -9 ; 5 \Big\} & &\end{align} \\ -----------------\)
  7. \(-8x=-\frac{1}{2}x^2-\frac{63}{2} \\ \Leftrightarrow \frac{1}{2}x^2-8x+\frac{63}{2}=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-8x+\frac{63}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2-16x+63=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+63=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.63 & &\\ & = 256-252 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-16)-\sqrt4}{2.1} & & = \frac{-(-16)+\sqrt4}{2.1} \\ & = \frac{14}{2} & & = \frac{18}{2} \\ & = 7 & & = 9 \\ \\ V &= \Big\{ 7 ; 9 \Big\} & &\end{align} \\ -----------------\)
  8. \(x(4x+4)=(x+1) \\ \Leftrightarrow 4x^2+4x=x+1 \\ \Leftrightarrow 4x^2+3x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+3x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.4.(-1) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.4} & & = \frac{-3+\sqrt25}{2.4} \\ & = \frac{-8}{8} & & = \frac{2}{8} \\ & = -1 & & = \frac{1}{4} \\ \\ V &= \Big\{ -1 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
  9. \((-5x+3)(-x-4)-x(4x+22)=-52\\ \Leftrightarrow 5x^2+20x-3x-12 -4x^2-22x+52=0 \\ \Leftrightarrow x^2-14x+40=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.40 & &\\ & = 196-160 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-14)-\sqrt36}{2.1} & & = \frac{-(-14)+\sqrt36}{2.1} \\ & = \frac{8}{2} & & = \frac{20}{2} \\ & = 4 & & = 10 \\ \\ V &= \Big\{ 4 ; 10 \Big\} & &\end{align} \\ -----------------\)
  10. \(\frac{1}{4}x^2+\frac{13}{12}x+1=0\\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{4}x^2+\frac{13}{12}x+1\right)=0 \color{red}{.12} \\ \Leftrightarrow 3x^2+13x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+13x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.3.12 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.3} & & = \frac{-13+\sqrt25}{2.3} \\ & = \frac{-18}{6} & & = \frac{-8}{6} \\ & = -3 & & = \frac{-4}{3} \\ \\ V &= \Big\{ -3 ; \frac{-4}{3} \Big\} & &\end{align} \\ -----------------\)
  11. \(\frac{1}{48}x^2-\frac{1}{4}x+\frac{3}{4}=0\\ \Leftrightarrow \color{red}{48.} \left(\frac{1}{48}x^2-\frac{1}{4}x+\frac{3}{4}\right)=0 \color{red}{.48} \\ \Leftrightarrow 4x^2-48x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-48x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-48)^2-4.4.144 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-48)}{2.4} & & \\ & = 6 & & \\V &= \Big\{ 6 \Big\} & &\end{align} \\ -----------------\)
  12. \((-3x+4)(4x+2)-x(-21x+74)=-136\\ \Leftrightarrow -12x^2-6x+16x+8 +21x^2-74x+136=0 \\ \Leftrightarrow 9x^2-72x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-72x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-72)^2-4.9.144 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-72)}{2.9} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
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