Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(9x^2+27x+48=-9x+12\)
2. \(4x^2-2x+90=8x-10\)
3. \(x^2+7x+91=-12x+7\)
4. \(9x^2+17x+7=12x+11\)
5. \(x^2+9x+47=-4x+7\)
6. \(16x^2+64x+64=0\)
7. \(72x^2+25x+2=0\)
8. \(x^2-17x+56=-3x+7\)
9. \(18x^2+25x+8=0\)
10. \(9x^2-20x+25=0\)
11. \(16x^2-34x+17=6x-8\)
12. \(2x^2+18x+20=5x+2\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(9x^2+27x+48=-9x+12\\ \Leftrightarrow 9x^2+36x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+36x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (36)^2-4.9.36 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-36}{2.9} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
2. \(4x^2-2x+90=8x-10\\ \Leftrightarrow 4x^2-10x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-10x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.4.100 & &\\ & = 100-1600 & & \\ & = -1500 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
3. \(x^2+7x+91=-12x+7\\ \Leftrightarrow x^2+19x+84=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+19x+84=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (19)^2-4.1.84 & &\\ & = 361-336 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-19-\sqrt25}{2.1} & & = \frac{-19+\sqrt25}{2.1} \\ & = \frac{-24}{2} & & = \frac{-14}{2} \\ & = -12 & & = -7 \\ \\ V &= \Big\{ -12 ; -7 \Big\} & &\end{align} \\ -----------------\)
4. \(9x^2+17x+7=12x+11\\ \Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.9.(-4) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\ & = \frac{-18}{18} & & = \frac{8}{18} \\ & = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)
5. \(x^2+9x+47=-4x+7\\ \Leftrightarrow x^2+13x+40=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+13x+40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.1.40 & &\\ & = 169-160 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt9}{2.1} & & = \frac{-13+\sqrt9}{2.1} \\ & = \frac{-16}{2} & & = \frac{-10}{2} \\ & = -8 & & = -5 \\ \\ V &= \Big\{ -8 ; -5 \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2+64x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (64)^2-4.16.64 & &\\ & = 4096-4096 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-64}{2.16} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{72x^2+25x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.72.2 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.72} & & = \frac{-25+\sqrt49}{2.72} \\ & = \frac{-32}{144} & & = \frac{-18}{144} \\ & = \frac{-2}{9} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
8. \(x^2-17x+56=-3x+7\\ \Leftrightarrow x^2-14x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.49 & &\\ & = 196-196 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-14)}{2.1} & & \\ & = 7 & & \\V &= \Big\{ 7 \Big\} & &\end{align} \\ -----------------\)
9. \(\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(\text{We zoeken de oplossingen van } \color{blue}{9x^2-20x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-20)^2-4.9.25 & &\\ & = 400-900 & & \\ & = -500 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
11. \(16x^2-34x+17=6x-8\\ \Leftrightarrow 16x^2-40x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-40x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-40)^2-4.16.25 & &\\ & = 1600-1600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-40)}{2.16} & & \\ & = \frac{5}{4} & & \\V &= \Big\{ \frac{5}{4} \Big\} & &\end{align} \\ -----------------\)
12. \(2x^2+18x+20=5x+2\\ \Leftrightarrow 2x^2+13x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+13x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.2.18 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.2} & & = \frac{-13+\sqrt25}{2.2} \\ & = \frac{-18}{4} & & = \frac{-8}{4} \\ & = \frac{-9}{2} & & = -2 \\ \\ V &= \Big\{ \frac{-9}{2} ; -2 \Big\} & &\end{align} \\ -----------------\)