Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(12x^2+2x-5=-5x+7\)
2. \(16x^2-10x+64=0\)
3. \(x^2-9x+18=0\)
4. \(x^2-14x+96=5x+6\)
5. \(4x^2+16x+16=0\)
6. \(48x^2+34x-3=9x-6\)
7. \(x^2-4x-77=0\)
8. \(12x^2+9x+7=4x+10\)
9. \(x^2-2x-80=0\)
10. \(x^2+29x+111=7x-10\)
11. \(x^2+2x-3=0\)
12. \(8x^2+14x+11=-11x-7\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(12x^2+2x-5=-5x+7\\ \Leftrightarrow 12x^2+7x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+7x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.12.(-12) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.12} & & = \frac{-7+\sqrt625}{2.12} \\ & = \frac{-32}{24} & & = \frac{18}{24} \\ & = \frac{-4}{3} & & = \frac{3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
2. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2-10x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.16.64 & &\\ & = 100-4096 & & \\ & = -3996 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
3. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-9x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-9)^2-4.1.18 & &\\ & = 81-72 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-9)-\sqrt9}{2.1} & & = \frac{-(-9)+\sqrt9}{2.1} \\ & = \frac{6}{2} & & = \frac{12}{2} \\ & = 3 & & = 6 \\ \\ V &= \Big\{ 3 ; 6 \Big\} & &\end{align} \\ -----------------\)
4. \(x^2-14x+96=5x+6\\ \Leftrightarrow x^2-19x+90=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-19x+90=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-19)^2-4.1.90 & &\\ & = 361-360 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-19)-\sqrt1}{2.1} & & = \frac{-(-19)+\sqrt1}{2.1} \\ & = \frac{18}{2} & & = \frac{20}{2} \\ & = 9 & & = 10 \\ \\ V &= \Big\{ 9 ; 10 \Big\} & &\end{align} \\ -----------------\)
5. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2+16x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.4.16 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.4} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
6. \(48x^2+34x-3=9x-6\\ \Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.48.3 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\ & = \frac{-32}{96} & & = \frac{-18}{96} \\ & = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-77=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-77) & &\\ & = 16+308 & & \\ & = 324 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt324}{2.1} & & = \frac{-(-4)+\sqrt324}{2.1} \\ & = \frac{-14}{2} & & = \frac{22}{2} \\ & = -7 & & = 11 \\ \\ V &= \Big\{ -7 ; 11 \Big\} & &\end{align} \\ -----------------\)
8. \(12x^2+9x+7=4x+10\\ \Leftrightarrow 12x^2+5x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+5x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.12.(-3) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.12} & & = \frac{-5+\sqrt169}{2.12} \\ & = \frac{-18}{24} & & = \frac{8}{24} \\ & = \frac{-3}{4} & & = \frac{1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
9. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-80=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-80) & &\\ & = 4+320 & & \\ & = 324 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt324}{2.1} & & = \frac{-(-2)+\sqrt324}{2.1} \\ & = \frac{-16}{2} & & = \frac{20}{2} \\ & = -8 & & = 10 \\ \\ V &= \Big\{ -8 ; 10 \Big\} & &\end{align} \\ -----------------\)
10. \(x^2+29x+111=7x-10\\ \Leftrightarrow x^2+22x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+22x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (22)^2-4.1.121 & &\\ & = 484-484 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-22}{2.1} & & \\ & = -11 & & \\V &= \Big\{ -11 \Big\} & &\end{align} \\ -----------------\)
11. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-3) & &\\ & = 4+12 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt16}{2.1} & & = \frac{-2+\sqrt16}{2.1} \\ & = \frac{-6}{2} & & = \frac{2}{2} \\ & = -3 & & = 1 \\ \\ V &= \Big\{ -3 ; 1 \Big\} & &\end{align} \\ -----------------\)
12. \(8x^2+14x+11=-11x-7\\ \Leftrightarrow 8x^2+25x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+25x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.8.18 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.8} & & = \frac{-25+\sqrt49}{2.8} \\ & = \frac{-32}{16} & & = \frac{-18}{16} \\ & = -2 & & = \frac{-9}{8} \\ \\ V &= \Big\{ -2 ; \frac{-9}{8} \Big\} & &\end{align} \\ -----------------\)