Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(2x^2+17x-26=12x-8\)
  2. \(x^2+15x+54=0\)
  3. \(9x^2+16x+32=6x-4\)
  4. \(16x^2-40x+25=0\)
  5. \(x^2-3x+19=7x+3\)
  6. \(9x^2-34x+44=8x-5\)
  7. \(9x^2+13x+3=7x+2\)
  8. \(2x^2+5x+2=0\)
  9. \(x^2+0x-12=6x+4\)
  10. \(4x^2-5x-2=x-11\)
  11. \(x^2+12x+130=8x+9\)
  12. \(x^2+7x-18=0\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(2x^2+17x-26=12x-8\\ \Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.(-18) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\ & = \frac{-18}{4} & & = \frac{8}{4} \\ & = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
  2. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+15x+54=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.1.54 & &\\ & = 225-216 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt9}{2.1} & & = \frac{-15+\sqrt9}{2.1} \\ & = \frac{-18}{2} & & = \frac{-12}{2} \\ & = -9 & & = -6 \\ \\ V &= \Big\{ -9 ; -6 \Big\} & &\end{align} \\ -----------------\)
  3. \(9x^2+16x+32=6x-4\\ \Leftrightarrow 9x^2+10x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+10x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.9.36 & &\\ & = 100-1296 & & \\ & = -1196 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  4. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2-40x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-40)^2-4.16.25 & &\\ & = 1600-1600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-40)}{2.16} & & \\ & = \frac{5}{4} & & \\V &= \Big\{ \frac{5}{4} \Big\} & &\end{align} \\ -----------------\)
  5. \(x^2-3x+19=7x+3\\ \Leftrightarrow x^2-10x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.16 & &\\ & = 100-64 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-10)-\sqrt36}{2.1} & & = \frac{-(-10)+\sqrt36}{2.1} \\ & = \frac{4}{2} & & = \frac{16}{2} \\ & = 2 & & = 8 \\ \\ V &= \Big\{ 2 ; 8 \Big\} & &\end{align} \\ -----------------\)
  6. \(9x^2-34x+44=8x-5\\ \Leftrightarrow 9x^2-42x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-42x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-42)^2-4.9.49 & &\\ & = 1764-1764 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-42)}{2.9} & & \\ & = \frac{7}{3} & & \\V &= \Big\{ \frac{7}{3} \Big\} & &\end{align} \\ -----------------\)
  7. \(9x^2+13x+3=7x+2\\ \Leftrightarrow 9x^2+6x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+6x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.9.1 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-6}{2.9} & & \\ & = -\frac{1}{3} & & \\V &= \Big\{ -\frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
  8. \(\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
  9. \(x^2+0x-12=6x+4\\ \Leftrightarrow x^2-6x-16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.(-16) & &\\ & = 36+64 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt100}{2.1} & & = \frac{-(-6)+\sqrt100}{2.1} \\ & = \frac{-4}{2} & & = \frac{16}{2} \\ & = -2 & & = 8 \\ \\ V &= \Big\{ -2 ; 8 \Big\} & &\end{align} \\ -----------------\)
  10. \(4x^2-5x-2=x-11\\ \Leftrightarrow 4x^2-6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.4.9 & &\\ & = 36-144 & & \\ & = -108 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  11. \(x^2+12x+130=8x+9\\ \Leftrightarrow x^2+4x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.121 & &\\ & = 16-484 & & \\ & = -468 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  12. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.(-18) & &\\ & = 49+72 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt121}{2.1} & & = \frac{-7+\sqrt121}{2.1} \\ & = \frac{-18}{2} & & = \frac{4}{2} \\ & = -9 & & = 2 \\ \\ V &= \Big\{ -9 ; 2 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator vanhoeckes.be/wiskunde 2022-08-18 11:58:48