Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(16x^2-17x+7=-11x+6\)
2. \(2x^2+13x+18=0\)
3. \(x^2-12x+32=0\)
4. \(x^2-9x-9=-10x+11\)
5. \(12x^2+17x+8=-8x-4\)
6. \(x^2+19x+43=6x+7\)
7. \(x^2-8x+7=0\)
8. \(9x^2+60x+100=0\)
9. \(18x^2+0x+5=-5x+7\)
10. \(x^2+11x+18=0\)
11. \(16x^2+6x+13=-2x+12\)
12. \(4x^2+34x+27=10x-9\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(16x^2-17x+7=-11x+6\\ \Leftrightarrow 16x^2-6x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-6x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.16.1 & &\\ & = 36-64 & & \\ & = -28 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
2. \(\text{We zoeken de oplossingen van } \color{blue}{2x^2+13x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.2.18 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.2} & & = \frac{-13+\sqrt25}{2.2} \\ & = \frac{-18}{4} & & = \frac{-8}{4} \\ & = \frac{-9}{2} & & = -2 \\ \\ V &= \Big\{ \frac{-9}{2} ; -2 \Big\} & &\end{align} \\ -----------------\)
3. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+32=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.32 & &\\ & = 144-128 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-12)-\sqrt16}{2.1} & & = \frac{-(-12)+\sqrt16}{2.1} \\ & = \frac{8}{2} & & = \frac{16}{2} \\ & = 4 & & = 8 \\ \\ V &= \Big\{ 4 ; 8 \Big\} & &\end{align} \\ -----------------\)
4. \(x^2-9x-9=-10x+11\\ \Leftrightarrow x^2+x-20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-20) & &\\ & = 1+80 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt81}{2.1} & & = \frac{-1+\sqrt81}{2.1} \\ & = \frac{-10}{2} & & = \frac{8}{2} \\ & = -5 & & = 4 \\ \\ V &= \Big\{ -5 ; 4 \Big\} & &\end{align} \\ -----------------\)
5. \(12x^2+17x+8=-8x-4\\ \Leftrightarrow 12x^2+25x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+25x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.12.12 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.12} & & = \frac{-25+\sqrt49}{2.12} \\ & = \frac{-32}{24} & & = \frac{-18}{24} \\ & = \frac{-4}{3} & & = \frac{-3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{-3}{4} \Big\} & &\end{align} \\ -----------------\)
6. \(x^2+19x+43=6x+7\\ \Leftrightarrow x^2+13x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+13x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.1.36 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.1} & & = \frac{-13+\sqrt25}{2.1} \\ & = \frac{-18}{2} & & = \frac{-8}{2} \\ & = -9 & & = -4 \\ \\ V &= \Big\{ -9 ; -4 \Big\} & &\end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+7=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.7 & &\\ & = 64-28 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt36}{2.1} & & = \frac{-(-8)+\sqrt36}{2.1} \\ & = \frac{2}{2} & & = \frac{14}{2} \\ & = 1 & & = 7 \\ \\ V &= \Big\{ 1 ; 7 \Big\} & &\end{align} \\ -----------------\)
8. \(\text{We zoeken de oplossingen van } \color{blue}{9x^2+60x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (60)^2-4.9.100 & &\\ & = 3600-3600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-60}{2.9} & & \\ & = -\frac{10}{3} & & \\V &= \Big\{ -\frac{10}{3} \Big\} & &\end{align} \\ -----------------\)
9. \(18x^2+0x+5=-5x+7\\ \Leftrightarrow 18x^2+5x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+5x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.18.(-2) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.18} & & = \frac{-5+\sqrt169}{2.18} \\ & = \frac{-18}{36} & & = \frac{8}{36} \\ & = \frac{-1}{2} & & = \frac{2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{2}{9} \Big\} & &\end{align} \\ -----------------\)
10. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.18 & &\\ & = 121-72 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt49}{2.1} & & = \frac{-11+\sqrt49}{2.1} \\ & = \frac{-18}{2} & & = \frac{-4}{2} \\ & = -9 & & = -2 \\ \\ V &= \Big\{ -9 ; -2 \Big\} & &\end{align} \\ -----------------\)
11. \(16x^2+6x+13=-2x+12\\ \Leftrightarrow 16x^2+8x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+8x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.16.1 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-8}{2.16} & & \\ & = -\frac{1}{4} & & \\V &= \Big\{ -\frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
12. \(4x^2+34x+27=10x-9\\ \Leftrightarrow 4x^2+24x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+24x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.4.36 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.4} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)