Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x^2+4x+18=-6x+2\)
2. \(4x^2+40x+100=0\)
3. \(16x^2+60x+111=4x-10\)
4. \(9x^2+36x+36=0\)
5. \(x^2+2x-20=-x+8\)
6. \(x^2-4x-77=0\)
7. \(6x^2+5x-6=0\)
8. \(8x^2+7x-18=0\)
9. \(24x^2-5x+0=-12x+6\)
10. \(16x^2+4x-4=-8x-8\)
11. \(9x^2-72x+144=0\)
12. \(36x^2+11x-14=4x-10\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x^2+4x+18=-6x+2\\ \Leftrightarrow x^2+10x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+10x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.1.16 & &\\ & = 100-64 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt36}{2.1} & & = \frac{-10+\sqrt36}{2.1} \\ & = \frac{-16}{2} & & = \frac{-4}{2} \\ & = -8 & & = -2 \\ \\ V &= \Big\{ -8 ; -2 \Big\} & &\end{align} \\ -----------------\)
2. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2+40x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (40)^2-4.4.100 & &\\ & = 1600-1600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-40}{2.4} & & \\ & = -5 & & \\V &= \Big\{ -5 \Big\} & &\end{align} \\ -----------------\)
3. \(16x^2+60x+111=4x-10\\ \Leftrightarrow 16x^2+56x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+56x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (56)^2-4.16.121 & &\\ & = 3136-7744 & & \\ & = -4608 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
4. \(\text{We zoeken de oplossingen van } \color{blue}{9x^2+36x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (36)^2-4.9.36 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-36}{2.9} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
5. \(x^2+2x-20=-x+8\\ \Leftrightarrow x^2+3x-28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-28=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-28) & &\\ & = 9+112 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt121}{2.1} & & = \frac{-3+\sqrt121}{2.1} \\ & = \frac{-14}{2} & & = \frac{8}{2} \\ & = -7 & & = 4 \\ \\ V &= \Big\{ -7 ; 4 \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-77=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-77) & &\\ & = 16+308 & & \\ & = 324 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt324}{2.1} & & = \frac{-(-4)+\sqrt324}{2.1} \\ & = \frac{-14}{2} & & = \frac{22}{2} \\ & = -7 & & = 11 \\ \\ V &= \Big\{ -7 ; 11 \Big\} & &\end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
8. \(\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.8.(-18) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\ & = \frac{-32}{16} & & = \frac{18}{16} \\ & = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)
9. \(24x^2-5x+0=-12x+6\\ \Leftrightarrow 24x^2+7x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+7x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.24.(-6) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.24} & & = \frac{-7+\sqrt625}{2.24} \\ & = \frac{-32}{48} & & = \frac{18}{48} \\ & = \frac{-2}{3} & & = \frac{3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{3}{8} \Big\} & &\end{align} \\ -----------------\)
10. \(16x^2+4x-4=-8x-8\\ \Leftrightarrow 16x^2+12x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+12x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.16.4 & &\\ & = 144-256 & & \\ & = -112 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
11. \(\text{We zoeken de oplossingen van } \color{blue}{9x^2-72x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-72)^2-4.9.144 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-72)}{2.9} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
12. \(36x^2+11x-14=4x-10\\ \Leftrightarrow 36x^2+7x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+7x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.36.(-4) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.36} & & = \frac{-7+\sqrt625}{2.36} \\ & = \frac{-32}{72} & & = \frac{18}{72} \\ & = \frac{-4}{9} & & = \frac{1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)