Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(16x^2-12x+7=6x-2\)
2. \(2x^2+19x+7=6x-11\)
3. \(9x^2-54x+81=0\)
4. \(x^2+22x+54=6x-9\)
5. \(x^2-22x+66=-6x+2\)
6. \(x^2+0x+12=-10x+3\)
7. \(x^2-10x+25=0\)
8. \(3x^2+11x-44=4x+4\)
9. \(4x^2+15x-4=0\)
10. \(x^2+7x+12=0\)
11. \(x^2-18x+72=0\)
12. \(4x^2-2x+9=0\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(16x^2-12x+7=6x-2\\ \Leftrightarrow 16x^2-18x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-18x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-18)^2-4.16.9 & &\\ & = 324-576 & & \\ & = -252 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
2. \(2x^2+19x+7=6x-11\\ \Leftrightarrow 2x^2+13x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+13x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.2.18 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.2} & & = \frac{-13+\sqrt25}{2.2} \\ & = \frac{-18}{4} & & = \frac{-8}{4} \\ & = \frac{-9}{2} & & = -2 \\ \\ V &= \Big\{ \frac{-9}{2} ; -2 \Big\} & &\end{align} \\ -----------------\)
3. \(\text{We zoeken de oplossingen van } \color{blue}{9x^2-54x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-54)^2-4.9.81 & &\\ & = 2916-2916 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-54)}{2.9} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
4. \(x^2+22x+54=6x-9\\ \Leftrightarrow x^2+16x+63=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+63=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.63 & &\\ & = 256-252 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-16-\sqrt4}{2.1} & & = \frac{-16+\sqrt4}{2.1} \\ & = \frac{-18}{2} & & = \frac{-14}{2} \\ & = -9 & & = -7 \\ \\ V &= \Big\{ -9 ; -7 \Big\} & &\end{align} \\ -----------------\)
5. \(x^2-22x+66=-6x+2\\ \Leftrightarrow x^2-16x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.64 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.1} & & \\ & = 8 & & \\V &= \Big\{ 8 \Big\} & &\end{align} \\ -----------------\)
6. \(x^2+0x+12=-10x+3\\ \Leftrightarrow x^2+10x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+10x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.1.9 & &\\ & = 100-36 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt64}{2.1} & & = \frac{-10+\sqrt64}{2.1} \\ & = \frac{-18}{2} & & = \frac{-2}{2} \\ & = -9 & & = -1 \\ \\ V &= \Big\{ -9 ; -1 \Big\} & &\end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-10x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.25 & &\\ & = 100-100 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-10)}{2.1} & & \\ & = 5 & & \\V &= \Big\{ 5 \Big\} & &\end{align} \\ -----------------\)
8. \(3x^2+11x-44=4x+4\\ \Leftrightarrow 3x^2+7x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+7x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.3.(-48) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.3} & & = \frac{-7+\sqrt625}{2.3} \\ & = \frac{-32}{6} & & = \frac{18}{6} \\ & = \frac{-16}{3} & & = 3 \\ \\ V &= \Big\{ \frac{-16}{3} ; 3 \Big\} & &\end{align} \\ -----------------\)
9. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.4.(-4) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\ & = \frac{-32}{8} & & = \frac{2}{8} \\ & = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
10. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+7x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.12 & &\\ & = 49-48 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt1}{2.1} & & = \frac{-7+\sqrt1}{2.1} \\ & = \frac{-8}{2} & & = \frac{-6}{2} \\ & = -4 & & = -3 \\ \\ V &= \Big\{ -4 ; -3 \Big\} & &\end{align} \\ -----------------\)
11. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-18x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-18)^2-4.1.72 & &\\ & = 324-288 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-18)-\sqrt36}{2.1} & & = \frac{-(-18)+\sqrt36}{2.1} \\ & = \frac{12}{2} & & = \frac{24}{2} \\ & = 6 & & = 12 \\ \\ V &= \Big\{ 6 ; 12 \Big\} & &\end{align} \\ -----------------\)
12. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2-2x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.4.9 & &\\ & = 4-144 & & \\ & = -140 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)