Vierkantsvergelijkingen (VKV)

Hoofdmenu Eentje per keer  🖨

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(9x^2+10x+4=0\)
2. \(18x^2+34x+3=9x-5\)
3. \(18x^2+13x+2=0\)
4. \(16x^2+6x+9=0\)
5. \(x^2+7x+6=0\)
6. \(x^2+x-2=0\)
7. \(48x^2+7x-3=0\)
8. \(4x^2+17x+4=0\)
9. \(x^2+11x+10=0\)
10. \(x^2+13x+36=0\)
11. \(x^2+17x+40=5x+4\)
12. \(9x^2+28x+81=0\)

---

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\text{We zoeken de oplossingen van } \color{blue}{9x^2+10x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.9.4 & &\\ & = 100-144 & & \\ & = -44 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
2. \(18x^2+34x+3=9x-5\\ \Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
3. \(\text{We zoeken de oplossingen van } \color{blue}{18x^2+13x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.18.2 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.18} & & = \frac{-13+\sqrt25}{2.18} \\ & = \frac{-18}{36} & & = \frac{-8}{36} \\ & = \frac{-1}{2} & & = \frac{-2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-2}{9} \Big\} & &\end{align} \\ -----------------\)
4. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2+6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.16.9 & &\\ & = 36-576 & & \\ & = -540 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
5. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+7x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.6 & &\\ & = 49-24 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt25}{2.1} & & = \frac{-7+\sqrt25}{2.1} \\ & = \frac{-12}{2} & & = \frac{-2}{2} \\ & = -6 & & = -1 \\ \\ V &= \Big\{ -6 ; -1 \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-2) & &\\ & = 1+8 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt9}{2.1} & & = \frac{-1+\sqrt9}{2.1} \\ & = \frac{-4}{2} & & = \frac{2}{2} \\ & = -2 & & = 1 \\ \\ V &= \Big\{ -2 ; 1 \Big\} & &\end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
8. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2+17x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.4.4 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.4} & & = \frac{-17+\sqrt225}{2.4} \\ & = \frac{-32}{8} & & = \frac{-2}{8} \\ & = -4 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -4 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
9. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.10 & &\\ & = 121-40 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt81}{2.1} & & = \frac{-11+\sqrt81}{2.1} \\ & = \frac{-20}{2} & & = \frac{-2}{2} \\ & = -10 & & = -1 \\ \\ V &= \Big\{ -10 ; -1 \Big\} & &\end{align} \\ -----------------\)
10. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+13x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.1.36 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.1} & & = \frac{-13+\sqrt25}{2.1} \\ & = \frac{-18}{2} & & = \frac{-8}{2} \\ & = -9 & & = -4 \\ \\ V &= \Big\{ -9 ; -4 \Big\} & &\end{align} \\ -----------------\)
11. \(x^2+17x+40=5x+4\\ \Leftrightarrow x^2+12x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.36 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-12}{2.1} & & \\ & = -6 & & \\V &= \Big\{ -6 \Big\} & &\end{align} \\ -----------------\)
12. \(\text{We zoeken de oplossingen van } \color{blue}{9x^2+28x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (28)^2-4.9.81 & &\\ & = 784-2916 & & \\ & = -2132 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)