Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(12x^2+32x+14=7x+2\)
2. \(4x^2-4x-4=-7x-3\)
3. \(x^2-11x+16=-9x+12\)
4. \(x^2-9x+37=-11x+12\)
5. \(x^2-23x+30=-11x-6\)
6. \(16x^2-62x+59=2x-5\)
7. \(36x^2+25x+4=0\)
8. \(x^2-6x-6=-11x-10\)
9. \(2x^2+37x+75=12x+3\)
10. \(16x^2+27x-8=12x-7\)
11. \(x^2-4x-12=0\)
12. \(x^2+2x+26=-9x+8\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(12x^2+32x+14=7x+2\\ \Leftrightarrow 12x^2+25x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+25x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.12.12 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.12} & & = \frac{-25+\sqrt49}{2.12} \\ & = \frac{-32}{24} & & = \frac{-18}{24} \\ & = \frac{-4}{3} & & = \frac{-3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{-3}{4} \Big\} & &\end{align} \\ -----------------\)
2. \(4x^2-4x-4=-7x-3\\ \Leftrightarrow 4x^2+3x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+3x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.4.(-1) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.4} & & = \frac{-3+\sqrt25}{2.4} \\ & = \frac{-8}{8} & & = \frac{2}{8} \\ & = -1 & & = \frac{1}{4} \\ \\ V &= \Big\{ -1 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
3. \(x^2-11x+16=-9x+12\\ \Leftrightarrow x^2-2x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.4 & &\\ & = 4-16 & & \\ & = -12 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
4. \(x^2-9x+37=-11x+12\\ \Leftrightarrow x^2+2x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.25 & &\\ & = 4-100 & & \\ & = -96 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
5. \(x^2-23x+30=-11x-6\\ \Leftrightarrow x^2-12x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.36 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-12)}{2.1} & & \\ & = 6 & & \\V &= \Big\{ 6 \Big\} & &\end{align} \\ -----------------\)
6. \(16x^2-62x+59=2x-5\\ \Leftrightarrow 16x^2-64x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-64x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-64)^2-4.16.64 & &\\ & = 4096-4096 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-64)}{2.16} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
8. \(x^2-6x-6=-11x-10\\ \Leftrightarrow x^2+5x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.4 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.1} & & = \frac{-5+\sqrt9}{2.1} \\ & = \frac{-8}{2} & & = \frac{-2}{2} \\ & = -4 & & = -1 \\ \\ V &= \Big\{ -4 ; -1 \Big\} & &\end{align} \\ -----------------\)
9. \(2x^2+37x+75=12x+3\\ \Leftrightarrow 2x^2+25x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+25x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.2.72 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.2} & & = \frac{-25+\sqrt49}{2.2} \\ & = \frac{-32}{4} & & = \frac{-18}{4} \\ & = -8 & & = \frac{-9}{2} \\ \\ V &= \Big\{ -8 ; \frac{-9}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(16x^2+27x-8=12x-7\\ \Leftrightarrow 16x^2+15x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+15x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.16.(-1) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.16} & & = \frac{-15+\sqrt289}{2.16} \\ & = \frac{-32}{32} & & = \frac{2}{32} \\ & = -1 & & = \frac{1}{16} \\ \\ V &= \Big\{ -1 ; \frac{1}{16} \Big\} & &\end{align} \\ -----------------\)
11. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-12) & &\\ & = 16+48 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt64}{2.1} & & = \frac{-(-4)+\sqrt64}{2.1} \\ & = \frac{-4}{2} & & = \frac{12}{2} \\ & = -2 & & = 6 \\ \\ V &= \Big\{ -2 ; 6 \Big\} & &\end{align} \\ -----------------\)
12. \(x^2+2x+26=-9x+8\\ \Leftrightarrow x^2+11x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.18 & &\\ & = 121-72 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt49}{2.1} & & = \frac{-11+\sqrt49}{2.1} \\ & = \frac{-18}{2} & & = \frac{-4}{2} \\ & = -9 & & = -2 \\ \\ V &= \Big\{ -9 ; -2 \Big\} & &\end{align} \\ -----------------\)