Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x^2-7x-49=-8x+7\)
2. \(6x^2+24x+14=-x-10\)
3. \(9x^2-45x+57=-7x-7\)
4. \(12x^2+13x+3=0\)
5. \(9x^2+22x+36=0\)
6. \(4x^2-16x+16=0\)
7. \(x^2-7x-8=0\)
8. \(x^2-13x+2=-6x-4\)
9. \(3x^2+13x+12=0\)
10. \(x^2-3x-18=0\)
11. \(x^2+5x-36=0\)
12. \(x^2+15x+54=0\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x^2-7x-49=-8x+7\\ \Leftrightarrow x^2+x-56=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-56=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-56) & &\\ & = 1+224 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt225}{2.1} & & = \frac{-1+\sqrt225}{2.1} \\ & = \frac{-16}{2} & & = \frac{14}{2} \\ & = -8 & & = 7 \\ \\ V &= \Big\{ -8 ; 7 \Big\} & &\end{align} \\ -----------------\)
2. \(6x^2+24x+14=-x-10\\ \Leftrightarrow 6x^2+25x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+25x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.6.24 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.6} & & = \frac{-25+\sqrt49}{2.6} \\ & = \frac{-32}{12} & & = \frac{-18}{12} \\ & = \frac{-8}{3} & & = \frac{-3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{-3}{2} \Big\} & &\end{align} \\ -----------------\)
3. \(9x^2-45x+57=-7x-7\\ \Leftrightarrow 9x^2-38x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-38x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-38)^2-4.9.64 & &\\ & = 1444-2304 & & \\ & = -860 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
4. \(\text{We zoeken de oplossingen van } \color{blue}{12x^2+13x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.12.3 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.12} & & = \frac{-13+\sqrt25}{2.12} \\ & = \frac{-18}{24} & & = \frac{-8}{24} \\ & = \frac{-3}{4} & & = \frac{-1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{-1}{3} \Big\} & &\end{align} \\ -----------------\)
5. \(\text{We zoeken de oplossingen van } \color{blue}{9x^2+22x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (22)^2-4.9.36 & &\\ & = 484-1296 & & \\ & = -812 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2-16x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.4.16 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.4} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.(-8) & &\\ & = 49+32 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt81}{2.1} & & = \frac{-(-7)+\sqrt81}{2.1} \\ & = \frac{-2}{2} & & = \frac{16}{2} \\ & = -1 & & = 8 \\ \\ V &= \Big\{ -1 ; 8 \Big\} & &\end{align} \\ -----------------\)
8. \(x^2-13x+2=-6x-4\\ \Leftrightarrow x^2-7x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.6 & &\\ & = 49-24 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt25}{2.1} & & = \frac{-(-7)+\sqrt25}{2.1} \\ & = \frac{2}{2} & & = \frac{12}{2} \\ & = 1 & & = 6 \\ \\ V &= \Big\{ 1 ; 6 \Big\} & &\end{align} \\ -----------------\)
9. \(\text{We zoeken de oplossingen van } \color{blue}{3x^2+13x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.3.12 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.3} & & = \frac{-13+\sqrt25}{2.3} \\ & = \frac{-18}{6} & & = \frac{-8}{6} \\ & = -3 & & = \frac{-4}{3} \\ \\ V &= \Big\{ -3 ; \frac{-4}{3} \Big\} & &\end{align} \\ -----------------\)
10. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-18) & &\\ & = 9+72 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt81}{2.1} & & = \frac{-(-3)+\sqrt81}{2.1} \\ & = \frac{-6}{2} & & = \frac{12}{2} \\ & = -3 & & = 6 \\ \\ V &= \Big\{ -3 ; 6 \Big\} & &\end{align} \\ -----------------\)
11. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-36) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.1} & & = \frac{-5+\sqrt169}{2.1} \\ & = \frac{-18}{2} & & = \frac{8}{2} \\ & = -9 & & = 4 \\ \\ V &= \Big\{ -9 ; 4 \Big\} & &\end{align} \\ -----------------\)
12. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+15x+54=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.1.54 & &\\ & = 225-216 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt9}{2.1} & & = \frac{-15+\sqrt9}{2.1} \\ & = \frac{-18}{2} & & = \frac{-12}{2} \\ & = -9 & & = -6 \\ \\ V &= \Big\{ -9 ; -6 \Big\} & &\end{align} \\ -----------------\)