Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(8x^2+16x+6=-x+4\)
2. \(4x^2+2x+17=-8x-8\)
3. \(36x^2+25x+4=0\)
4. \(16x^2+16x+4=0\)
5. \(x^2+8x+15=0\)
6. \(4x^2+20x+25=0\)
7. \(16x^2+46x+60=8x-4\)
8. \(x^2-21x+110=0\)
9. \(x^2-10x-24=0\)
10. \(x^2-10x-11=0\)
11. \(4x^2+14x+36=0\)
12. \(x^2-4x-18=-9x-4\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(8x^2+16x+6=-x+4\\ \Leftrightarrow 8x^2+17x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+17x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.8.2 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.8} & & = \frac{-17+\sqrt225}{2.8} \\ & = \frac{-32}{16} & & = \frac{-2}{16} \\ & = -2 & & = \frac{-1}{8} \\ \\ V &= \Big\{ -2 ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
2. \(4x^2+2x+17=-8x-8\\ \Leftrightarrow 4x^2+10x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+10x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.4.25 & &\\ & = 100-400 & & \\ & = -300 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
3. \(\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
4. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2+16x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.16.4 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.16} & & \\ & = -\frac{1}{2} & & \\V &= \Big\{ -\frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
5. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+8x+15=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.15 & &\\ & = 64-60 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-8-\sqrt4}{2.1} & & = \frac{-8+\sqrt4}{2.1} \\ & = \frac{-10}{2} & & = \frac{-6}{2} \\ & = -5 & & = -3 \\ \\ V &= \Big\{ -5 ; -3 \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2+20x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.4.25 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.4} & & \\ & = -\frac{5}{2} & & \\V &= \Big\{ -\frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
7. \(16x^2+46x+60=8x-4\\ \Leftrightarrow 16x^2+38x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+38x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (38)^2-4.16.64 & &\\ & = 1444-4096 & & \\ & = -2652 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
8. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-21x+110=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-21)^2-4.1.110 & &\\ & = 441-440 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-21)-\sqrt1}{2.1} & & = \frac{-(-21)+\sqrt1}{2.1} \\ & = \frac{20}{2} & & = \frac{22}{2} \\ & = 10 & & = 11 \\ \\ V &= \Big\{ 10 ; 11 \Big\} & &\end{align} \\ -----------------\)
9. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-10x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.(-24) & &\\ & = 100+96 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-10)-\sqrt196}{2.1} & & = \frac{-(-10)+\sqrt196}{2.1} \\ & = \frac{-4}{2} & & = \frac{24}{2} \\ & = -2 & & = 12 \\ \\ V &= \Big\{ -2 ; 12 \Big\} & &\end{align} \\ -----------------\)
10. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-10x-11=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.(-11) & &\\ & = 100+44 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-10)-\sqrt144}{2.1} & & = \frac{-(-10)+\sqrt144}{2.1} \\ & = \frac{-2}{2} & & = \frac{22}{2} \\ & = -1 & & = 11 \\ \\ V &= \Big\{ -1 ; 11 \Big\} & &\end{align} \\ -----------------\)
11. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2+14x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.4.36 & &\\ & = 196-576 & & \\ & = -380 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
12. \(x^2-4x-18=-9x-4\\ \Leftrightarrow x^2+5x-14=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-14=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-14) & &\\ & = 25+56 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt81}{2.1} & & = \frac{-5+\sqrt81}{2.1} \\ & = \frac{-14}{2} & & = \frac{4}{2} \\ & = -7 & & = 2 \\ \\ V &= \Big\{ -7 ; 2 \Big\} & &\end{align} \\ -----------------\)