Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(2x^2+20x+14=7x-4\)
2. \(9x^2+14x+123=2x+2\)
3. \(x^2-15x+37=-3x+2\)
4. \(x^2+16x+64=0\)
5. \(36x^2+18x+0=-7x-4\)
6. \(4x^2-2x+144=0\)
7. \(16x^2-48x+36=0\)
8. \(x^2+5x+6=9x+2\)
9. \(4x^2+4x+1=0\)
10. \(12x^2-3x-13=-8x-10\)
11. \(8x^2+12x-23=5x-5\)
12. \(16x^2+x+9=-9x+8\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(2x^2+20x+14=7x-4\\ \Leftrightarrow 2x^2+13x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+13x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.2.18 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.2} & & = \frac{-13+\sqrt25}{2.2} \\ & = \frac{-18}{4} & & = \frac{-8}{4} \\ & = \frac{-9}{2} & & = -2 \\ \\ V &= \Big\{ \frac{-9}{2} ; -2 \Big\} & &\end{align} \\ -----------------\)
2. \(9x^2+14x+123=2x+2\\ \Leftrightarrow 9x^2+12x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+12x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.9.121 & &\\ & = 144-4356 & & \\ & = -4212 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
3. \(x^2-15x+37=-3x+2\\ \Leftrightarrow x^2-12x+35=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+35=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.35 & &\\ & = 144-140 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-12)-\sqrt4}{2.1} & & = \frac{-(-12)+\sqrt4}{2.1} \\ & = \frac{10}{2} & & = \frac{14}{2} \\ & = 5 & & = 7 \\ \\ V &= \Big\{ 5 ; 7 \Big\} & &\end{align} \\ -----------------\)
4. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.64 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.1} & & \\ & = -8 & & \\V &= \Big\{ -8 \Big\} & &\end{align} \\ -----------------\)
5. \(36x^2+18x+0=-7x-4\\ \Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2-2x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.4.144 & &\\ & = 4-2304 & & \\ & = -2300 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2-48x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-48)^2-4.16.36 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-48)}{2.16} & & \\ & = \frac{3}{2} & & \\V &= \Big\{ \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
8. \(x^2+5x+6=9x+2\\ \Leftrightarrow x^2-4x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-4)}{2.1} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
9. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2+4x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.4.1 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.4} & & \\ & = -\frac{1}{2} & & \\V &= \Big\{ -\frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(12x^2-3x-13=-8x-10\\ \Leftrightarrow 12x^2+5x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+5x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.12.(-3) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.12} & & = \frac{-5+\sqrt169}{2.12} \\ & = \frac{-18}{24} & & = \frac{8}{24} \\ & = \frac{-3}{4} & & = \frac{1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
11. \(8x^2+12x-23=5x-5\\ \Leftrightarrow 8x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.8.(-18) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\ & = \frac{-32}{16} & & = \frac{18}{16} \\ & = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)
12. \(16x^2+x+9=-9x+8\\ \Leftrightarrow 16x^2+10x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+10x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.16.1 & &\\ & = 100-64 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt36}{2.16} & & = \frac{-10+\sqrt36}{2.16} \\ & = \frac{-16}{32} & & = \frac{-4}{32} \\ & = \frac{-1}{2} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)