Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x^2-8x+54=-12x+5\)
2. \(9x^2-12x+6=-4x+2\)
3. \(x^2+10x-106=8x-7\)
4. \(x^2+6x+5=0\)
5. \(4x^2+18x+6=2x-10\)
6. \(x^2+3x-70=0\)
7. \(x^2-x-12=0\)
8. \(18x^2+7x-8=0\)
9. \(x^2+5x-24=3x-9\)
10. \(2x^2+0x+10=-3x+12\)
11. \(x^2-5x-14=0\)
12. \(x^2-4x+4=0\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x^2-8x+54=-12x+5\\ \Leftrightarrow x^2+4x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.49 & &\\ & = 16-196 & & \\ & = -180 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
2. \(9x^2-12x+6=-4x+2\\ \Leftrightarrow 9x^2-8x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-8x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.9.4 & &\\ & = 64-144 & & \\ & = -80 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
3. \(x^2+10x-106=8x-7\\ \Leftrightarrow x^2+2x-99=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-99=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-99) & &\\ & = 4+396 & & \\ & = 400 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt400}{2.1} & & = \frac{-2+\sqrt400}{2.1} \\ & = \frac{-22}{2} & & = \frac{18}{2} \\ & = -11 & & = 9 \\ \\ V &= \Big\{ -11 ; 9 \Big\} & &\end{align} \\ -----------------\)
4. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+5=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.5 & &\\ & = 36-20 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt16}{2.1} & & = \frac{-6+\sqrt16}{2.1} \\ & = \frac{-10}{2} & & = \frac{-2}{2} \\ & = -5 & & = -1 \\ \\ V &= \Big\{ -5 ; -1 \Big\} & &\end{align} \\ -----------------\)
5. \(4x^2+18x+6=2x-10\\ \Leftrightarrow 4x^2+16x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+16x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.4.16 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.4} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-70=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-70) & &\\ & = 9+280 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt289}{2.1} & & = \frac{-3+\sqrt289}{2.1} \\ & = \frac{-20}{2} & & = \frac{14}{2} \\ & = -10 & & = 7 \\ \\ V &= \Big\{ -10 ; 7 \Big\} & &\end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-12) & &\\ & = 1+48 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt49}{2.1} & & = \frac{-(-1)+\sqrt49}{2.1} \\ & = \frac{-6}{2} & & = \frac{8}{2} \\ & = -3 & & = 4 \\ \\ V &= \Big\{ -3 ; 4 \Big\} & &\end{align} \\ -----------------\)
8. \(\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
9. \(x^2+5x-24=3x-9\\ \Leftrightarrow x^2+2x-15=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-15=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-15) & &\\ & = 4+60 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt64}{2.1} & & = \frac{-2+\sqrt64}{2.1} \\ & = \frac{-10}{2} & & = \frac{6}{2} \\ & = -5 & & = 3 \\ \\ V &= \Big\{ -5 ; 3 \Big\} & &\end{align} \\ -----------------\)
10. \(2x^2+0x+10=-3x+12\\ \Leftrightarrow 2x^2+3x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+3x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.2.(-2) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.2} & & = \frac{-3+\sqrt25}{2.2} \\ & = \frac{-8}{4} & & = \frac{2}{4} \\ & = -2 & & = \frac{1}{2} \\ \\ V &= \Big\{ -2 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
11. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-14=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-14) & &\\ & = 25+56 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt81}{2.1} & & = \frac{-(-5)+\sqrt81}{2.1} \\ & = \frac{-4}{2} & & = \frac{14}{2} \\ & = -2 & & = 7 \\ \\ V &= \Big\{ -2 ; 7 \Big\} & &\end{align} \\ -----------------\)
12. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-4)}{2.1} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)