Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(3x^2+0x-59=-7x-11\)
2. \(12x^2+13x+3=0\)
3. \(x^2-4x+3=0\)
4. \(x^2-15x+36=0\)
5. \(x^2-4x+6=-11x-4\)
6. \(9x^2-66x+121=0\)
7. \(x^2+8x+81=0\)
8. \(16x^2-64x+87=8x+6\)
9. \(6x^2-4x-20=-11x+4\)
10. \(16x^2+8x+1=-7x+2\)
11. \(x^2-8x-53=-9x+3\)
12. \(x^2+17x+72=0\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(3x^2+0x-59=-7x-11\\ \Leftrightarrow 3x^2+7x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+7x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.3.(-48) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.3} & & = \frac{-7+\sqrt625}{2.3} \\ & = \frac{-32}{6} & & = \frac{18}{6} \\ & = \frac{-16}{3} & & = 3 \\ \\ V &= \Big\{ \frac{-16}{3} ; 3 \Big\} & &\end{align} \\ -----------------\)
2. \(\text{We zoeken de oplossingen van } \color{blue}{12x^2+13x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.12.3 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.12} & & = \frac{-13+\sqrt25}{2.12} \\ & = \frac{-18}{24} & & = \frac{-8}{24} \\ & = \frac{-3}{4} & & = \frac{-1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{-1}{3} \Big\} & &\end{align} \\ -----------------\)
3. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.3 & &\\ & = 16-12 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt4}{2.1} & & = \frac{-(-4)+\sqrt4}{2.1} \\ & = \frac{2}{2} & & = \frac{6}{2} \\ & = 1 & & = 3 \\ \\ V &= \Big\{ 1 ; 3 \Big\} & &\end{align} \\ -----------------\)
4. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-15x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-15)^2-4.1.36 & &\\ & = 225-144 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-15)-\sqrt81}{2.1} & & = \frac{-(-15)+\sqrt81}{2.1} \\ & = \frac{6}{2} & & = \frac{24}{2} \\ & = 3 & & = 12 \\ \\ V &= \Big\{ 3 ; 12 \Big\} & &\end{align} \\ -----------------\)
5. \(x^2-4x+6=-11x-4\\ \Leftrightarrow x^2+7x+10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+7x+10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.10 & &\\ & = 49-40 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt9}{2.1} & & = \frac{-7+\sqrt9}{2.1} \\ & = \frac{-10}{2} & & = \frac{-4}{2} \\ & = -5 & & = -2 \\ \\ V &= \Big\{ -5 ; -2 \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{9x^2-66x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-66)^2-4.9.121 & &\\ & = 4356-4356 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-66)}{2.9} & & \\ & = \frac{11}{3} & & \\V &= \Big\{ \frac{11}{3} \Big\} & &\end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+8x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.81 & &\\ & = 64-324 & & \\ & = -260 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
8. \(16x^2-64x+87=8x+6\\ \Leftrightarrow 16x^2-72x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-72x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-72)^2-4.16.81 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-72)}{2.16} & & \\ & = \frac{9}{4} & & \\V &= \Big\{ \frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
9. \(6x^2-4x-20=-11x+4\\ \Leftrightarrow 6x^2+7x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+7x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.6.(-24) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.6} & & = \frac{-7+\sqrt625}{2.6} \\ & = \frac{-32}{12} & & = \frac{18}{12} \\ & = \frac{-8}{3} & & = \frac{3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(16x^2+8x+1=-7x+2\\ \Leftrightarrow 16x^2+15x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+15x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.16.(-1) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.16} & & = \frac{-15+\sqrt289}{2.16} \\ & = \frac{-32}{32} & & = \frac{2}{32} \\ & = -1 & & = \frac{1}{16} \\ \\ V &= \Big\{ -1 ; \frac{1}{16} \Big\} & &\end{align} \\ -----------------\)
11. \(x^2-8x-53=-9x+3\\ \Leftrightarrow x^2+x-56=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-56=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-56) & &\\ & = 1+224 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt225}{2.1} & & = \frac{-1+\sqrt225}{2.1} \\ & = \frac{-16}{2} & & = \frac{14}{2} \\ & = -8 & & = 7 \\ \\ V &= \Big\{ -8 ; 7 \Big\} & &\end{align} \\ -----------------\)
12. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+17x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.1.72 & &\\ & = 289-288 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt1}{2.1} & & = \frac{-17+\sqrt1}{2.1} \\ & = \frac{-18}{2} & & = \frac{-16}{2} \\ & = -9 & & = -8 \\ \\ V &= \Big\{ -9 ; -8 \Big\} & &\end{align} \\ -----------------\)