Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x^2+14x+45=0\)
2. \(3x^2+17x+56=-8x+8\)
3. \(x^2+5x+14=x+10\)
4. \(16x^2+30x+55=-4x-9\)
5. \(24x^2+24x-1=-x-7\)
6. \(16x^2-48x+36=0\)
7. \(16x^2+88x+121=0\)
8. \(18x^2-4x+2=-11x+10\)
9. \(x^2+18x+80=0\)
10. \(16x^2-5x+2=11x-2\)
11. \(x^2-8x-48=0\)
12. \(x^2+17x-72=10x-12\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+45=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.45 & &\\ & = 196-180 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-14-\sqrt16}{2.1} & & = \frac{-14+\sqrt16}{2.1} \\ & = \frac{-18}{2} & & = \frac{-10}{2} \\ & = -9 & & = -5 \\ \\ V &= \Big\{ -9 ; -5 \Big\} & &\end{align} \\ -----------------\)
2. \(3x^2+17x+56=-8x+8\\ \Leftrightarrow 3x^2+25x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+25x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.3.48 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.3} & & = \frac{-25+\sqrt49}{2.3} \\ & = \frac{-32}{6} & & = \frac{-18}{6} \\ & = \frac{-16}{3} & & = -3 \\ \\ V &= \Big\{ \frac{-16}{3} ; -3 \Big\} & &\end{align} \\ -----------------\)
3. \(x^2+5x+14=x+10\\ \Leftrightarrow x^2+4x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.1} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
4. \(16x^2+30x+55=-4x-9\\ \Leftrightarrow 16x^2+34x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+34x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (34)^2-4.16.64 & &\\ & = 1156-4096 & & \\ & = -2940 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
5. \(24x^2+24x-1=-x-7\\ \Leftrightarrow 24x^2+25x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+25x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.24.6 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.24} & & = \frac{-25+\sqrt49}{2.24} \\ & = \frac{-32}{48} & & = \frac{-18}{48} \\ & = \frac{-2}{3} & & = \frac{-3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{-3}{8} \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2-48x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-48)^2-4.16.36 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-48)}{2.16} & & \\ & = \frac{3}{2} & & \\V &= \Big\{ \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2+88x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (88)^2-4.16.121 & &\\ & = 7744-7744 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-88}{2.16} & & \\ & = -\frac{11}{4} & & \\V &= \Big\{ -\frac{11}{4} \Big\} & &\end{align} \\ -----------------\)
8. \(18x^2-4x+2=-11x+10\\ \Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
9. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+18x+80=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (18)^2-4.1.80 & &\\ & = 324-320 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-18-\sqrt4}{2.1} & & = \frac{-18+\sqrt4}{2.1} \\ & = \frac{-20}{2} & & = \frac{-16}{2} \\ & = -10 & & = -8 \\ \\ V &= \Big\{ -10 ; -8 \Big\} & &\end{align} \\ -----------------\)
10. \(16x^2-5x+2=11x-2\\ \Leftrightarrow 16x^2-16x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-16x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.16.4 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.16} & & \\ & = \frac{1}{2} & & \\V &= \Big\{ \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
11. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-48) & &\\ & = 64+192 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt256}{2.1} & & = \frac{-(-8)+\sqrt256}{2.1} \\ & = \frac{-8}{2} & & = \frac{24}{2} \\ & = -4 & & = 12 \\ \\ V &= \Big\{ -4 ; 12 \Big\} & &\end{align} \\ -----------------\)
12. \(x^2+17x-72=10x-12\\ \Leftrightarrow x^2+7x-60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+7x-60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.(-60) & &\\ & = 49+240 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt289}{2.1} & & = \frac{-7+\sqrt289}{2.1} \\ & = \frac{-24}{2} & & = \frac{10}{2} \\ & = -12 & & = 5 \\ \\ V &= \Big\{ -12 ; 5 \Big\} & &\end{align} \\ -----------------\)