Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(4x^2-32x+64=0\)
2. \(48x^2+7x-3=0\)
3. \(x^2-4x+4=0\)
4. \(x^2-2x-47=6x+1\)
5. \(2x^2+19x+30=6x+12\)
6. \(8x^2+27x+2=12x+4\)
7. \(x^2-21x+91=-3x+10\)
8. \(x^2+7x+151=-3x+7\)
9. \(x^2-8x+15=0\)
10. \(9x^2+22x+4=-2x-12\)
11. \(3x^2+5x-12=0\)
12. \(9x^2-4x+4=-9x+8\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2-32x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-32)^2-4.4.64 & &\\ & = 1024-1024 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-32)}{2.4} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
2. \(\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
3. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-4)}{2.1} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
4. \(x^2-2x-47=6x+1\\ \Leftrightarrow x^2-8x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-48) & &\\ & = 64+192 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt256}{2.1} & & = \frac{-(-8)+\sqrt256}{2.1} \\ & = \frac{-8}{2} & & = \frac{24}{2} \\ & = -4 & & = 12 \\ \\ V &= \Big\{ -4 ; 12 \Big\} & &\end{align} \\ -----------------\)
5. \(2x^2+19x+30=6x+12\\ \Leftrightarrow 2x^2+13x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+13x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.2.18 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.2} & & = \frac{-13+\sqrt25}{2.2} \\ & = \frac{-18}{4} & & = \frac{-8}{4} \\ & = \frac{-9}{2} & & = -2 \\ \\ V &= \Big\{ \frac{-9}{2} ; -2 \Big\} & &\end{align} \\ -----------------\)
6. \(8x^2+27x+2=12x+4\\ \Leftrightarrow 8x^2+15x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+15x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.8.(-2) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.8} & & = \frac{-15+\sqrt289}{2.8} \\ & = \frac{-32}{16} & & = \frac{2}{16} \\ & = -2 & & = \frac{1}{8} \\ \\ V &= \Big\{ -2 ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
7. \(x^2-21x+91=-3x+10\\ \Leftrightarrow x^2-18x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-18x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-18)^2-4.1.81 & &\\ & = 324-324 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-18)}{2.1} & & \\ & = 9 & & \\V &= \Big\{ 9 \Big\} & &\end{align} \\ -----------------\)
8. \(x^2+7x+151=-3x+7\\ \Leftrightarrow x^2+10x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+10x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.1.144 & &\\ & = 100-576 & & \\ & = -476 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
9. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+15=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.15 & &\\ & = 64-60 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt4}{2.1} & & = \frac{-(-8)+\sqrt4}{2.1} \\ & = \frac{6}{2} & & = \frac{10}{2} \\ & = 3 & & = 5 \\ \\ V &= \Big\{ 3 ; 5 \Big\} & &\end{align} \\ -----------------\)
10. \(9x^2+22x+4=-2x-12\\ \Leftrightarrow 9x^2+24x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+24x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.9.16 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.9} & & \\ & = -\frac{4}{3} & & \\V &= \Big\{ -\frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
11. \(\text{We zoeken de oplossingen van } \color{blue}{3x^2+5x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.3.(-12) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.3} & & = \frac{-5+\sqrt169}{2.3} \\ & = \frac{-18}{6} & & = \frac{8}{6} \\ & = -3 & & = \frac{4}{3} \\ \\ V &= \Big\{ -3 ; \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
12. \(9x^2-4x+4=-9x+8\\ \Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.9.(-4) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\ & = \frac{-18}{18} & & = \frac{8}{18} \\ & = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)