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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(\frac{25}{9}x=-\frac{1}{3}x^2-\frac{16}{3}\)
  2. \(x(6x+11)=6(x+1)\)
  3. \(2x^2-(10x+33)=x(x-2)\)
  4. \((-4x+4)(x+5)-x(-13x+24)=4\)
  5. \(x(x+13)=15(x+1)\)
  6. \(-(13-27x)=-4x^2-(49-2x)\)
  7. \(-2x=-\frac{3}{5}x^2-\frac{5}{3}\)
  8. \(\frac{1}{48}x^2+\frac{1}{4}x+\frac{3}{4}=0\)
  9. \((4x+1)(-5x+1)-x(-21x+18)=-35\)
  10. \(-(7-28x)=-x^2-(97-9x)\)
  11. \(x(16x+6)=-4(x+1)\)
  12. \(20x^2-(9x-4)=11x(x-2)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(\frac{25}{9}x=-\frac{1}{3}x^2-\frac{16}{3} \\ \Leftrightarrow \frac{1}{3}x^2+\frac{25}{9}x+\frac{16}{3}=0 \\ \Leftrightarrow \color{red}{9.} \left(\frac{1}{3}x^2+\frac{25}{9}x+\frac{16}{3}\right)=0 \color{red}{.9} \\ \Leftrightarrow 3x^2+25x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+25x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.3.48 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.3} & & = \frac{-25+\sqrt49}{2.3} \\ & = \frac{-32}{6} & & = \frac{-18}{6} \\ & = \frac{-16}{3} & & = -3 \\ \\ V &= \Big\{ \frac{-16}{3} ; -3 \Big\} & &\end{align} \\ -----------------\)
  2. \(x(6x+11)=6(x+1) \\ \Leftrightarrow 6x^2+11x=6x+6 \\ \Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
  3. \(2x^2-(10x+33)=x(x-2) \\ \Leftrightarrow 2x^2-10x-33=x^2-2x \\ \Leftrightarrow x^2-8x-33=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-33=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-33) & &\\ & = 64+132 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt196}{2.1} & & = \frac{-(-8)+\sqrt196}{2.1} \\ & = \frac{-6}{2} & & = \frac{22}{2} \\ & = -3 & & = 11 \\ \\ V &= \Big\{ -3 ; 11 \Big\} & &\end{align} \\ -----------------\)
  4. \((-4x+4)(x+5)-x(-13x+24)=4\\ \Leftrightarrow -4x^2-20x+4x+20 +13x^2-24x-4=0 \\ \Leftrightarrow 9x^2-24x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-24x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.9.16 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-24)}{2.9} & & \\ & = \frac{4}{3} & & \\V &= \Big\{ \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
  5. \(x(x+13)=15(x+1) \\ \Leftrightarrow x^2+13x=15x+15 \\ \Leftrightarrow x^2-2x-15=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-15=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-15) & &\\ & = 4+60 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt64}{2.1} & & = \frac{-(-2)+\sqrt64}{2.1} \\ & = \frac{-6}{2} & & = \frac{10}{2} \\ & = -3 & & = 5 \\ \\ V &= \Big\{ -3 ; 5 \Big\} & &\end{align} \\ -----------------\)
  6. \(-(13-27x)=-4x^2-(49-2x) \\ \Leftrightarrow -13+27x=-4x^2-49+2x \\ \Leftrightarrow 4x^2+25x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+25x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.4.36 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.4} & & = \frac{-25+\sqrt49}{2.4} \\ & = \frac{-32}{8} & & = \frac{-18}{8} \\ & = -4 & & = \frac{-9}{4} \\ \\ V &= \Big\{ -4 ; \frac{-9}{4} \Big\} & &\end{align} \\ -----------------\)
  7. \(-2x=-\frac{3}{5}x^2-\frac{5}{3} \\ \Leftrightarrow \frac{3}{5}x^2-2x+\frac{5}{3}=0 \\ \Leftrightarrow \color{red}{15.} \left(\frac{3}{5}x^2-2x+\frac{5}{3}\right)=0 \color{red}{.15} \\ \Leftrightarrow 9x^2-30x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-30x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-30)^2-4.9.25 & &\\ & = 900-900 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-30)}{2.9} & & \\ & = \frac{5}{3} & & \\V &= \Big\{ \frac{5}{3} \Big\} & &\end{align} \\ -----------------\)
  8. \(\frac{1}{48}x^2+\frac{1}{4}x+\frac{3}{4}=0\\ \Leftrightarrow \color{red}{48.} \left(\frac{1}{48}x^2+\frac{1}{4}x+\frac{3}{4}\right)=0 \color{red}{.48} \\ \Leftrightarrow x^2+12x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.36 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-12}{2.1} & & \\ & = -6 & & \\V &= \Big\{ -6 \Big\} & &\end{align} \\ -----------------\)
  9. \((4x+1)(-5x+1)-x(-21x+18)=-35\\ \Leftrightarrow -20x^2+4x-5x+1 +21x^2-18x+35=0 \\ \Leftrightarrow x^2-13x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-13x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-13)^2-4.1.36 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-13)-\sqrt25}{2.1} & & = \frac{-(-13)+\sqrt25}{2.1} \\ & = \frac{8}{2} & & = \frac{18}{2} \\ & = 4 & & = 9 \\ \\ V &= \Big\{ 4 ; 9 \Big\} & &\end{align} \\ -----------------\)
  10. \(-(7-28x)=-x^2-(97-9x) \\ \Leftrightarrow -7+28x=-x^2-97+9x \\ \Leftrightarrow x^2+19x+90=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+19x+90=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (19)^2-4.1.90 & &\\ & = 361-360 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-19-\sqrt1}{2.1} & & = \frac{-19+\sqrt1}{2.1} \\ & = \frac{-20}{2} & & = \frac{-18}{2} \\ & = -10 & & = -9 \\ \\ V &= \Big\{ -10 ; -9 \Big\} & &\end{align} \\ -----------------\)
  11. \(x(16x+6)=-4(x+1) \\ \Leftrightarrow 16x^2+6x=-4x-4 \\ \Leftrightarrow 16x^2+10x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+10x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.16.4 & &\\ & = 100-256 & & \\ & = -156 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  12. \(20x^2-(9x-4)=11x(x-2) \\ \Leftrightarrow 20x^2-9x+4=11x^2-22x \\ \Leftrightarrow 9x^2+13x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+13x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.9.4 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.9} & & = \frac{-13+\sqrt25}{2.9} \\ & = \frac{-18}{18} & & = \frac{-8}{18} \\ & = -1 & & = \frac{-4}{9} \\ \\ V &= \Big\{ -1 ; \frac{-4}{9} \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator vanhoeckes.be/wiskunde 2021-12-02 02:01:24