Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(16x^2+50x+81=0\)
2. \(x^2-2x+11=-9x+1\)
3. \(x^2-13x-53=-12x+3\)
4. \(x^2-14x+118=8x-2\)
5. \(6x^2+5x-6=0\)
6. \(9x^2+5x-6=7x-7\)
7. \(x^2+7x-18=0\)
8. \(2x^2+19x-17=4x-9\)
9. \(16x^2+21x-9=11x-10\)
10. \(x^2+14x+49=0\)
11. \(x^2-2x+54=11x+12\)
12. \(x^2-17x+60=-3x+12\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2+50x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (50)^2-4.16.81 & &\\ & = 2500-5184 & & \\ & = -2684 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
2. \(x^2-2x+11=-9x+1\\ \Leftrightarrow x^2+7x+10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+7x+10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.10 & &\\ & = 49-40 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt9}{2.1} & & = \frac{-7+\sqrt9}{2.1} \\ & = \frac{-10}{2} & & = \frac{-4}{2} \\ & = -5 & & = -2 \\ \\ V &= \Big\{ -5 ; -2 \Big\} & &\end{align} \\ -----------------\)
3. \(x^2-13x-53=-12x+3\\ \Leftrightarrow x^2-x-56=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-56=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-56) & &\\ & = 1+224 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt225}{2.1} & & = \frac{-(-1)+\sqrt225}{2.1} \\ & = \frac{-14}{2} & & = \frac{16}{2} \\ & = -7 & & = 8 \\ \\ V &= \Big\{ -7 ; 8 \Big\} & &\end{align} \\ -----------------\)
4. \(x^2-14x+118=8x-2\\ \Leftrightarrow x^2-22x+120=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-22x+120=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.1.120 & &\\ & = 484-480 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-22)-\sqrt4}{2.1} & & = \frac{-(-22)+\sqrt4}{2.1} \\ & = \frac{20}{2} & & = \frac{24}{2} \\ & = 10 & & = 12 \\ \\ V &= \Big\{ 10 ; 12 \Big\} & &\end{align} \\ -----------------\)
5. \(\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
6. \(9x^2+5x-6=7x-7\\ \Leftrightarrow 9x^2-2x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-2x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.9.1 & &\\ & = 4-36 & & \\ & = -32 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.(-18) & &\\ & = 49+72 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt121}{2.1} & & = \frac{-7+\sqrt121}{2.1} \\ & = \frac{-18}{2} & & = \frac{4}{2} \\ & = -9 & & = 2 \\ \\ V &= \Big\{ -9 ; 2 \Big\} & &\end{align} \\ -----------------\)
8. \(2x^2+19x-17=4x-9\\ \Leftrightarrow 2x^2+15x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+15x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.2.(-8) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.2} & & = \frac{-15+\sqrt289}{2.2} \\ & = \frac{-32}{4} & & = \frac{2}{4} \\ & = -8 & & = \frac{1}{2} \\ \\ V &= \Big\{ -8 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
9. \(16x^2+21x-9=11x-10\\ \Leftrightarrow 16x^2+10x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+10x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.16.1 & &\\ & = 100-64 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt36}{2.16} & & = \frac{-10+\sqrt36}{2.16} \\ & = \frac{-16}{32} & & = \frac{-4}{32} \\ & = \frac{-1}{2} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
10. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.49 & &\\ & = 196-196 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-14}{2.1} & & \\ & = -7 & & \\V &= \Big\{ -7 \Big\} & &\end{align} \\ -----------------\)
11. \(x^2-2x+54=11x+12\\ \Leftrightarrow x^2-13x+42=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-13x+42=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-13)^2-4.1.42 & &\\ & = 169-168 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-13)-\sqrt1}{2.1} & & = \frac{-(-13)+\sqrt1}{2.1} \\ & = \frac{12}{2} & & = \frac{14}{2} \\ & = 6 & & = 7 \\ \\ V &= \Big\{ 6 ; 7 \Big\} & &\end{align} \\ -----------------\)
12. \(x^2-17x+60=-3x+12\\ \Leftrightarrow x^2-14x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.48 & &\\ & = 196-192 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-14)-\sqrt4}{2.1} & & = \frac{-(-14)+\sqrt4}{2.1} \\ & = \frac{12}{2} & & = \frac{16}{2} \\ & = 6 & & = 8 \\ \\ V &= \Big\{ 6 ; 8 \Big\} & &\end{align} \\ -----------------\)