Vierkantsvergelijkingen (VKV)

Hoofdmenu Eentje per keer  🖨

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(2x^2+7x-72=0\)
2. \(16x^2+18x+3=-6x-6\)
3. \(9x^2-54x+104=6x+4\)
4. \(18x^2+5x-2=0\)
5. \(36x^2+17x+0=10x+4\)
6. \(16x^2-8x+1=0\)
7. \(4x^2+5x+1=0\)
8. \(4x^2+2x-10=-2x-11\)
9. \(x^2-4x+41=10x-8\)
10. \(x^2+16x-83=11x+1\)
11. \(x^2-13x+19=-11x-6\)
12. \(x^2+4x-68=11x-8\)

---

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
2. \(16x^2+18x+3=-6x-6\\ \Leftrightarrow 16x^2+24x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+24x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.16.9 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.16} & & \\ & = -\frac{3}{4} & & \\V &= \Big\{ -\frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
3. \(9x^2-54x+104=6x+4\\ \Leftrightarrow 9x^2-60x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-60x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-60)^2-4.9.100 & &\\ & = 3600-3600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-60)}{2.9} & & \\ & = \frac{10}{3} & & \\V &= \Big\{ \frac{10}{3} \Big\} & &\end{align} \\ -----------------\)
4. \(\text{We zoeken de oplossingen van } \color{blue}{18x^2+5x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.18.(-2) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.18} & & = \frac{-5+\sqrt169}{2.18} \\ & = \frac{-18}{36} & & = \frac{8}{36} \\ & = \frac{-1}{2} & & = \frac{2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{2}{9} \Big\} & &\end{align} \\ -----------------\)
5. \(36x^2+17x+0=10x+4\\ \Leftrightarrow 36x^2+7x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+7x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.36.(-4) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.36} & & = \frac{-7+\sqrt625}{2.36} \\ & = \frac{-32}{72} & & = \frac{18}{72} \\ & = \frac{-4}{9} & & = \frac{1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2-8x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.16.1 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-8)}{2.16} & & \\ & = \frac{1}{4} & & \\V &= \Big\{ \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.1 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.4} & & = \frac{-5+\sqrt9}{2.4} \\ & = \frac{-8}{8} & & = \frac{-2}{8} \\ & = -1 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -1 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
8. \(4x^2+2x-10=-2x-11\\ \Leftrightarrow 4x^2+4x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+4x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.4.1 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.4} & & \\ & = -\frac{1}{2} & & \\V &= \Big\{ -\frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
9. \(x^2-4x+41=10x-8\\ \Leftrightarrow x^2-14x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.49 & &\\ & = 196-196 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-14)}{2.1} & & \\ & = 7 & & \\V &= \Big\{ 7 \Big\} & &\end{align} \\ -----------------\)
10. \(x^2+16x-83=11x+1\\ \Leftrightarrow x^2+5x-84=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-84=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-84) & &\\ & = 25+336 & & \\ & = 361 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt361}{2.1} & & = \frac{-5+\sqrt361}{2.1} \\ & = \frac{-24}{2} & & = \frac{14}{2} \\ & = -12 & & = 7 \\ \\ V &= \Big\{ -12 ; 7 \Big\} & &\end{align} \\ -----------------\)
11. \(x^2-13x+19=-11x-6\\ \Leftrightarrow x^2-2x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.25 & &\\ & = 4-100 & & \\ & = -96 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
12. \(x^2+4x-68=11x-8\\ \Leftrightarrow x^2-7x-60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x-60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.(-60) & &\\ & = 49+240 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt289}{2.1} & & = \frac{-(-7)+\sqrt289}{2.1} \\ & = \frac{-10}{2} & & = \frac{24}{2} \\ & = -5 & & = 12 \\ \\ V &= \Big\{ -5 ; 12 \Big\} & &\end{align} \\ -----------------\)