Vierkantsvergelijkingen (VKV)

Hoofdmenu Eentje per keer  🖨

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x^2-7x-8=0\)
2. \(x^2+7x-30=0\)
3. \(x^2-3x+20=8x-10\)
4. \(6x^2-7x+5=-12x+11\)
5. \(x^2+2x-48=0\)
6. \(x^2-11x+10=0\)
7. \(16x^2-29x+35=11x+10\)
8. \(9x^2+12x+6=7x+10\)
9. \(72x^2+15x+5=8x+7\)
10. \(8x^2+10x+6=-7x+4\)
11. \(16x^2+17x+1=0\)
12. \(x^2+11x+30=0\)

---

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.(-8) & &\\ & = 49+32 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt81}{2.1} & & = \frac{-(-7)+\sqrt81}{2.1} \\ & = \frac{-2}{2} & & = \frac{16}{2} \\ & = -1 & & = 8 \\ \\ V &= \Big\{ -1 ; 8 \Big\} & &\end{align} \\ -----------------\)
2. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+7x-30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.(-30) & &\\ & = 49+120 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt169}{2.1} & & = \frac{-7+\sqrt169}{2.1} \\ & = \frac{-20}{2} & & = \frac{6}{2} \\ & = -10 & & = 3 \\ \\ V &= \Big\{ -10 ; 3 \Big\} & &\end{align} \\ -----------------\)
3. \(x^2-3x+20=8x-10\\ \Leftrightarrow x^2-11x+30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-11)^2-4.1.30 & &\\ & = 121-120 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-11)-\sqrt1}{2.1} & & = \frac{-(-11)+\sqrt1}{2.1} \\ & = \frac{10}{2} & & = \frac{12}{2} \\ & = 5 & & = 6 \\ \\ V &= \Big\{ 5 ; 6 \Big\} & &\end{align} \\ -----------------\)
4. \(6x^2-7x+5=-12x+11\\ \Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
5. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-48) & &\\ & = 4+192 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt196}{2.1} & & = \frac{-2+\sqrt196}{2.1} \\ & = \frac{-16}{2} & & = \frac{12}{2} \\ & = -8 & & = 6 \\ \\ V &= \Big\{ -8 ; 6 \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-11)^2-4.1.10 & &\\ & = 121-40 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-11)-\sqrt81}{2.1} & & = \frac{-(-11)+\sqrt81}{2.1} \\ & = \frac{2}{2} & & = \frac{20}{2} \\ & = 1 & & = 10 \\ \\ V &= \Big\{ 1 ; 10 \Big\} & &\end{align} \\ -----------------\)
7. \(16x^2-29x+35=11x+10\\ \Leftrightarrow 16x^2-40x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-40x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-40)^2-4.16.25 & &\\ & = 1600-1600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-40)}{2.16} & & \\ & = \frac{5}{4} & & \\V &= \Big\{ \frac{5}{4} \Big\} & &\end{align} \\ -----------------\)
8. \(9x^2+12x+6=7x+10\\ \Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.9.(-4) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\ & = \frac{-18}{18} & & = \frac{8}{18} \\ & = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)
9. \(72x^2+15x+5=8x+7\\ \Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.72.(-2) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\ & = \frac{-32}{144} & & = \frac{18}{144} \\ & = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
10. \(8x^2+10x+6=-7x+4\\ \Leftrightarrow 8x^2+17x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+17x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.8.2 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.8} & & = \frac{-17+\sqrt225}{2.8} \\ & = \frac{-32}{16} & & = \frac{-2}{16} \\ & = -2 & & = \frac{-1}{8} \\ \\ V &= \Big\{ -2 ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
11. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2+17x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.16.1 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.16} & & = \frac{-17+\sqrt225}{2.16} \\ & = \frac{-32}{32} & & = \frac{-2}{32} \\ & = -1 & & = \frac{-1}{16} \\ \\ V &= \Big\{ -1 ; \frac{-1}{16} \Big\} & &\end{align} \\ -----------------\)
12. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.30 & &\\ & = 121-120 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt1}{2.1} & & = \frac{-11+\sqrt1}{2.1} \\ & = \frac{-12}{2} & & = \frac{-10}{2} \\ & = -6 & & = -5 \\ \\ V &= \Big\{ -6 ; -5 \Big\} & &\end{align} \\ -----------------\)