Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x^2+x-1=-2x+9\)
2. \(24x^2+35x+2=10x-4\)
3. \(4x^2-14x+8=-6x-1\)
4. \(x^2-6x+9=0\)
5. \(x^2-4x+4=0\)
6. \(x^2+16x+60=0\)
7. \(6x^2+4x+10=-9x+4\)
8. \(x^2+9x+14=0\)
9. \(4x^2+38x+139=-10x-5\)
10. \(x^2-x-2=0\)
11. \(48x^2+0x-13=-7x-10\)
12. \(x^2-18x+117=4x-3\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x^2+x-1=-2x+9\\ \Leftrightarrow x^2+3x-10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-10) & &\\ & = 9+40 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt49}{2.1} & & = \frac{-3+\sqrt49}{2.1} \\ & = \frac{-10}{2} & & = \frac{4}{2} \\ & = -5 & & = 2 \\ \\ V &= \Big\{ -5 ; 2 \Big\} & &\end{align} \\ -----------------\)
2. \(24x^2+35x+2=10x-4\\ \Leftrightarrow 24x^2+25x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+25x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.24.6 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.24} & & = \frac{-25+\sqrt49}{2.24} \\ & = \frac{-32}{48} & & = \frac{-18}{48} \\ & = \frac{-2}{3} & & = \frac{-3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{-3}{8} \Big\} & &\end{align} \\ -----------------\)
3. \(4x^2-14x+8=-6x-1\\ \Leftrightarrow 4x^2-8x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-8x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.4.9 & &\\ & = 64-144 & & \\ & = -80 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
4. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.9 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.1} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
5. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-4)}{2.1} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.60 & &\\ & = 256-240 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-16-\sqrt16}{2.1} & & = \frac{-16+\sqrt16}{2.1} \\ & = \frac{-20}{2} & & = \frac{-12}{2} \\ & = -10 & & = -6 \\ \\ V &= \Big\{ -10 ; -6 \Big\} & &\end{align} \\ -----------------\)
7. \(6x^2+4x+10=-9x+4\\ \Leftrightarrow 6x^2+13x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+13x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.6.6 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.6} & & = \frac{-13+\sqrt25}{2.6} \\ & = \frac{-18}{12} & & = \frac{-8}{12} \\ & = \frac{-3}{2} & & = \frac{-2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{-2}{3} \Big\} & &\end{align} \\ -----------------\)
8. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+9x+14=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (9)^2-4.1.14 & &\\ & = 81-56 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-9-\sqrt25}{2.1} & & = \frac{-9+\sqrt25}{2.1} \\ & = \frac{-14}{2} & & = \frac{-4}{2} \\ & = -7 & & = -2 \\ \\ V &= \Big\{ -7 ; -2 \Big\} & &\end{align} \\ -----------------\)
9. \(4x^2+38x+139=-10x-5\\ \Leftrightarrow 4x^2+48x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+48x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (48)^2-4.4.144 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-48}{2.4} & & \\ & = -6 & & \\V &= \Big\{ -6 \Big\} & &\end{align} \\ -----------------\)
10. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-2) & &\\ & = 1+8 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt9}{2.1} & & = \frac{-(-1)+\sqrt9}{2.1} \\ & = \frac{-2}{2} & & = \frac{4}{2} \\ & = -1 & & = 2 \\ \\ V &= \Big\{ -1 ; 2 \Big\} & &\end{align} \\ -----------------\)
11. \(48x^2+0x-13=-7x-10\\ \Leftrightarrow 48x^2+7x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
12. \(x^2-18x+117=4x-3\\ \Leftrightarrow x^2-22x+120=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-22x+120=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.1.120 & &\\ & = 484-480 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-22)-\sqrt4}{2.1} & & = \frac{-(-22)+\sqrt4}{2.1} \\ & = \frac{20}{2} & & = \frac{24}{2} \\ & = 10 & & = 12 \\ \\ V &= \Big\{ 10 ; 12 \Big\} & &\end{align} \\ -----------------\)